1.) Angel. His strength, motivation, and dedication to the mission made him an indespensible member. Angel is characteristically more pragmatic and logical then Buffy, and when it came to making the hard decisions, Angel wasn't afraid to do what had to be done. Angel, although an Informant to the Scoobies, was never really a member yet while on his own show he assumed a superior role as the Team's leader.
Angel's tough but let us be clear about this, Buffy is just as tough, and probably tougher. Additionally, while most people would agree that Angel does tend to be more logical and pragmatic then Buffy on most occasions, the Scoobies weren't just Buffy, they also had Giles. The Scoobies didn't need Buffy to be pragmatic they had Giles who had the Slayers ear.
Buffy is instinctual, and her instincts are good. When instructing the Potentials in Potential she tells them to use their instincts. This lesson serves them well. The Scoobs know this as well, and trust in it. This unifies the group. To quote Xander in the Body "We follow Buffy. It's what we do." They have faith in their leader. So if one is trying to say that Buffy can't make the tough call, check out 'The Gift' or 'Chosen' shes ready and willing to lay down peoples lives in pursuit of her goal, those lives including any number of Slayers, Anya, Xanders eye, herself etc.
Is Angel a good leader? It's debateable, he summarily fired his entire staff in S2 without so much as an explaination, that is not the mark of a good leader. So perhaps his pragmatism and logic get's in the way of being a truly inspiring leader that Buffy is. I would say that faith and friendship is far more effective and efficient a leadership quality then pragmatism and logic.
2.) That Vision Thing. Cordy's visions given to her from Doyle proved indespensible. The Scoobies had nothing of the sort, sans Buffy's prophetic dreams. For A.I., Cordy's visions enbabled the team to get a "head start" on crimes about to happen or in-progress, enabling them to save more people. Adversely, Buffy and her gang were always researching to learn more about something, and that could take days or even weeks. Innocent people surely died due to that fact that Buffy was simply not aware that they needed help.
One of the things we forget is that part of the Slayers power is that she receives visions in her dreams that instruct her in the ways to fight evil. Although not a direct link to the PTB it was a nice thing to have now and again. Not only that but the efficiency and effectiveness of those visions is not absolute. Cordelias visions have been hacked by W&H that caused Angel to release Billy. They caused her excruciating amounts of pain and physical injury so much so that she ended up making a deal with a devil that inadvertantly caused her to become the vessel for she who would rule the world and eat souls Jasmine. Between that and pouring through some dusty books I'll take Giles in my ear any day of the week.
It's also hard to argue about who and how many people did or did not die based on studying in libraries in Sunnydale. We could make the counter argument that while Angel was off doing the bidding of some questionable PtB that may or may not have been plotting to rule the world he could have been better used elsewhere and because he wasn't there many people died.
When you look closer, it's pretty hard to argue that Cordys visions made them significantly more efficient or effective then good ol' fashioned research.
3.) Advertising. Thanks to Cordy's initative, Angel Investigations had business cards and a website, which put the word out there that their services were available. In Sunnydale, a victim in need of help had to be lucky enough to be in trouble while Buffy was in the area, - but due to A.I.'s advertising and word-of-mouth reputation, customers knew who and where they could turn to for help.
The Scoobies were forced to keep their operations a secret; in part due to the fact that Buffy's identity as the Slayer needed to be kept a secret for her and her friends' protection. Of course, it was assumed that the residents of Sunnydale were unaware of the demonic activity surrounding the Hellmouth. A.I, on the other hand, did not feel the need to hide the fact that demons existed to their clients or correspondants.
One thing to remember is that Buffy's mission statement is not "Help the helpless". That's Angel Investigations. The mission statement of the slayer is "Into each generation a Slayer is born: one girl in all the world, a chosen one. She alone will wield the strength and skill to fight the vampires, demons, and the forces of darkness; to stop the spread of their evil and the swell of their numbers." Buffys mission was adopted by her Watcher and the rest of the Scoobies and you don't need business cards for that kind of business. Which team was more efficient and effective? I've already pointed that out in my opening statement. The Scoobies by the numbers come out quite ahead.
It is also debatable that their customers didn't know who she was. Some of her customers such as the entire '99 class of Seniors at Sunnydale High School seemed to know that she was something special as we see in 'The Prom.' Not only that but a number of other organizations knew where to turn for help such as the Watchers Countil, the Knights of Byzantium, the Monks of Dagon, the witch coven of Devon, the United States Department of Defense and more that I can't think of at the moment. As we see, the presence of demons, and humanities protectors was not as secret as argued in the sleepy town of Sunnydale.
There are also a number of fun anecdotes in regards to the utter ineffectiveness of A.I. advertisement. For instance 'Lonely Hearts' when Cordelias unabashed advertisement causes Doyle a lot of trouble when someone figures she must be a prostitute. How about when in 'Provider' when they passed out 6,000 flyers that had the wrong phone number on them. The Advertising was how they got their business to some degree, but it was neither that efficient or effective.
Advertising is nice, but clearly A.I. was not very efficient at it. I think that ultimately Buffys 'word of mouth' (Read: kicking ass completely for years on end) advertising was just as, if not more effective.
Angel Investigations was more effective due to the fact that they had more resources available to them. Buffy and the Scoobies were working out of a library with little resources and a lot of limitations, where A.I. were essentially free to do as they pleased in a professional manner.
It's not necessarily determined that A.I. had significantly more resources then the Scoobies. In the early seasons of both shows, they were little more then shoe string operations operating out of a school library for the former and a dungy office/delapitated 80 year old hotel for the latter. Later on when the Scoobs move their base of operations to the Magic Bag they become extremely effective at having the type of resources they need available, everything from a literal store full of magical items and references both mundane and rare and additionally as we see in OMWF, a fairly large and tastefully appointed dojo area in the back. Likewise as A.I. settles into the Hyperion they end up with a small dojo in the basement and a decent (although not as large as the Magic Bag) collection of books in Angel/Wes's office.
The only time the resources of each team wildly diverges is Buffy S7 versus Angel S5 and yes it's easy to see that 'Buffys home she can barely afford' might as well be a park bench and card board box compared to Angels 'branch of Hell Incorporated complete with tactical squads, a library full of infinate knowledge and enough cars to almost make Angel go Angelus on us' is that really a deal that comes out in Angels favor? In order to get these resources he had to make a deal that well... you've all watched S5 and know about After the Fall... ended up killing Wes, Gunn, Fred and sent everyone in L.A. to hell. That's not a very effective OR efficient deal.
As opposed to the residents of Sunnydale, the residents of L.A. had a greater awareness of demons in general. A.I, not having to hide their true intentions or operations allowed them to focus on more of the important things; fighting evil, and saving people - as Cordy coined their motto, "helping the hopeless".
Buffy's partial anonymity was by choice, and protected her from reprisals. Angel's was the anonymity of existing in the big city, and not being able to go out in the daytime. The latter was a liability. Buffy's anonymity was a strength. It did not prevent her from searching the areas she needed to, nor did it prevent access to any area of the town. Angel had less access to many parts of L.A. then Buffy had to Sunnydale.
It's also incorrect to say that the residents of L.A. were more greatly aware of anything. From season 1 with poor Kate Lockley being called 'Scully' by all her cop buddies because she dared to believe and all the people on the subway in 'The Prodigal' when all the people on the subway swear that a smelly crazy demon was a homeless guy on crack all the way to S5 'Unleashed' with Nina being blind sided by the craziness of the city or 'Origin' when Connors new parents report their sons surviving being run over by a car, the city of L.A. is not more keenly aware of anything.
The reason why there are more people in L.A. who seem to 'know whats going on' is larger only because it's a city of 12 million people, not a small suburban town, so even if the percentage of people who 'know' is the same, that's a heck of a lot more people in L.A. then Sunnydale.
As we can see, Angels lack of anonymity was neither more efficient nor more of an effective way to solve his goals then Buffy's relative secrecy, and if anything was a hinderence to him.
Yet perhaps the most important addition to A.I was Cordelia herself. It was her who convinced Angel and Doyle to make their agency "official" and to start charging clients for their services. The income brought in allowed them to expand and afford to hire Wes and then Gunn. And of course, it was her visions that gave Angel and his Team something tangible to fight for. Wesley, with his vast knowledge of demonolgy and expertise in the handling of weapons, became A.I.'s "Giles" figure, while Gunn as the "muscle" served some serious man-power. Angel the decision maker, and the woman who gives them visions to follow and fight, make A.I complete. The later additions and the name change to "Team Angel" only strengthen the group.
Commercializing AI was never successful enough to see it as a major strength. Staffing was never the Scoobs problem. In fact with Faith, Spike, Cordelia and even Angel, Scooby central shared many of the same staff AI did, and did it without financing. The one time AI did concentrate on commercial success in 'Provider', things fell apart. The conclusion in that episode was capitalist obsession was a bad choice for AI. Later with access to the fortunes of W & H they failed at their goals. Money is not a factor to success in the demon fighting biz. The Scoobs remained loyal, to duty, the mission and each other. Ultimately they were always more successful.
Another problem is that as you point out, they had to make money to hire on Wes so that he could become their 'Giles' figure. Well by comparison the Scoobs already had a 'Giles' figure, his name was Giles, and he didn't need to be hired or trained, he was already in place. Not exactly more efficient or effective to have to hire something the other team already has.
It is important to note that A.I. continued to function without Angel. When Buffy leaves at the end of Season 3, we see the Scoobies attempting to fight in her absence, but failing miserably. The difference is that A.I. was more of a team then the Scoobies were because every member was equal with a uniqiue skill. After Buffy dies in 'The Gift', we see the Scoobies yet again attempting to keep her legacy alive - now with the help of Spike and Anya - but still failing obviously without The Slayer and her Watcher.
In Season 2 when Buffy leaves it's true that the Scoobies struggle with fighting the Vamps. Although in the opening of S6 that their skills have significantly improved. What with the telepathic connections and Buffybot. They were effective enough that the 'outside world' believed that the Buffster was still protecting Sunnydale the entire summer. Yes, that didn't last, and an evil biker gang ran the place over, but it certainly was a better showing then the last time.
Contrast that with the fate of A.I. during Angels absences. In S2 when Angel fires everyone and they start their own A.I. is the first time. However as we see in 'Epiphany' without Angels timely arrival, Cordelia would have hatched an eyeball out of the back of her head and Wes and Gunn weren't far behind. They would have all died.
The second time that A.I. is without Angel for an extended period of time is in S4 when they bring back Angelus. That was a cluster if we ever saw one. While again the team did a better job, it was just shockingly chaotic. Fred and Gunn broke up, Gunn and Wes almost got in a fist fight, Connor tried to kill Angel, the entire city of L.A. descended into insanity, Lilah was killed, Cordelia was free to go crazy etc. etc. Bad times all around. Not only that but how did A.I. solve their missing leader problem? They called in a certain free agent Slayer and a certain lesbian Scoobie!
This brings up an interesting point, how often do the Scoobs have to bring in an a member of A.I. to help handle their problems? I'll wait for you to think about it.... yeah once, end of S7 when Angel brings that cheeky little medallion that burns Spike in a pillar of fire. By contrast Oz brings Angel the Gem of Amarra, Cordelia gets computer help from Willow a number of times, Willow saves the day in Orpheus, Willow was again attempted to be contacted in 'Shells' and Andrew assists in 'Damage.' For the record I don't count the various Buffy and Angel visits in S4/S1 as particularly helpful on either end heh.
If you look at the facts you can easily see two points. One, that either team without its leader is damaged and much less effective. Two, that over time the Scoobies have thrown a lot more helping hands A.I.s way then the other way around. In a debate over efficiency and effectiveness it seems that while neither team can handle losing their boss, when it does happen A.I. needs a lot more outside help.
A.I.'s connections and clients also helped them out in their business, which made them more efficient and successful. Buffy's only "inside" informants were Willy the Snitch and Spike - who both proved to be less than accurate sources of information, and whom certainly had an agenda. In other words, A.I. was more willing to accept help from outside sources, while Buffy had to for the most part rely on her own instincts and her immediate circle of friends for information.
Unlike the Scoobies, who were wary to add to their already close-knit group, Angel's Team were more than willing to accept new members into their group. The additions of Fred, Lorne, Spike, Conner, and Illyria increased the group's power. Also, A.I's use of Allies, such as Gwen, Nina, and Faith, strengthened their ability to accomplish their goals.
I don't think your argument regarding outside sources is necessarily very well supported. I think the Scoobs and A.I. both made good use of outside contacts. The Scoobs had everyone from Angel in S1, Willy, Spike, the Watchers Council from time to time, the U.S. Government in S4 and from time to time onward, the witches coven in England, Faith and I'm sure others I can't think of at the moment.
They also regularly added to their group, over time they added Cordelia, Oz, Wesley, Anya, Spike, Robin Wood, Tara, and Andrew. A.I. was very accepting of adding people as well there is no doubt, over time they added Wes, Fred, Gunn, Lorne, Illyria and Spike which is great, but I don't think either team was particularly hostile towards accepting new members.
Finally, let's not forget that certain outside contacts that A.I. made weren't so useful. For instance, Lilah and Lindsay could have towards the end of both their lives been considered allies. Neither of them had Angels best interest at heart. Additionally, we can always look back at his ally 'Wolfram & Hart' in S5 that led him towards a path that led to the destruction of all of L.A.
A careful examination of the situation shows that A.I. was no more efficient or effective when it came to using outside sources or accepting new members into their group. If anything the Scoobies were better judges of character and never allied with a person or organization that inadvertantly perverted their goals and turned them 'evil' like what happened to A.I.
And lastly, As a professional group that was more business-oriented, A.I. knew when to put their personal issues aside and do the job right. The Scoobies has a lot of history and personal drama which often conflicted with their roles as Buffy's assistants to her mission. With A.I, each member knew their place and their role and what was expected of them, therefore building a more dynamic, effective, and professional team.
This claim does not stand up well when examined closely either. Angel had a very hard time putting personal issues aside and doing the job right if you look at it. In S2 he fires his entire staff over personal issues with Darla. In S3 he allows himself to be distracted so thoroughly by Cordelia that he misses Wesleys obvious need for help which leads to Connorgate, him losing his best man, trying to MURDER his best friend, and a season and a half of estrangement. He allows his team to be co-opted by W&H, his primary nemesis, in order to protect his son and his love, letting personal issues again get in the way of doing the job.
Not to mention due to personal drama within the organization, Connor tried to muder Angel, Wesley almost killed Gunn at one point, Angel almost killed Wesley, Lorne had to kill Lindsay, Illyria at one point murdered every member single handedly and only was saved by a luck time jump, Spike and Angel nearly tore each others faces off going after a destiny, and I'm sure I've missed one or two.
Certainly the Scoobies had their own share of personal drama as well. Even Willow didn't show a strong desire to murder any of her friends during her S6 fall out. Even Anya when she went all Demony again didn't REALLY want to kill Buffy as they discussed at the end of their fight.
If you look at the facts you can see that personal issues regularly get in the way of both groups. However if you look at the effect if has on A.I. versus the Scoobies you can see that it's not very efficient or effective to have your group members attempting to murder each other, which A.I. seems to have a proclivity for.
It's important to point out that A.I. were in many ways awesome, but if you look at the various points raised by Joan, none of them show in any substantive way that A.I. was more efficient or effective then the Scoobies.
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Special props to Nerd4Hire and ILLYRIAN, they throw out the ideas, I'm just writing them down.
