No, they don't have that right. They may have certain skills that we don't, and they put their lives on the line to keep the world safe, but that still doesn't give them the right to dictate whether or not someone should be killed. Angel was all about atonement and redemption, yet he took that away from Lindsey with a second thought. That's hardly fair.
Also, Angel signed away the Shanshu. I haven't read the Angel comics, so I don't know what's happened there, but at the time of NFA, he'd signed away his 'claim' to the Shanshu, so that argument doesn't hold water for me.
Lindsey was given a chance twice. That's more than enough. It's not one of those addict rehabilitation situations wherein the addict slips up and does drugs, even just a little amount. It's more like, a choice on a game show like Deal or No Deal. When you make the wrong choice, you loose the money. If you made the right one, you get the jackpot.
Angel only did that to convince the Circle. It's not that he didn't care about the prophecy. He had faith/hope about the prophecy ever since "The Cautionary tale of Numero Cinco". Besides, it happened on the same day. Even if it he were no longer "the vampire" of the Sanshu, it was just a few hours.
Besides, in the comic, the Sanshu was signed off bu Angel but it was not yet "Filed" as an official document because W&H had an Earthquake and it probably got lost. Not to mention, everyone in the Blackthorn that could have "filed" it the next day were dead on the same night.
Buffy doesn't shrug off killing people though. Ever. She took her place as a leader because she had to, and we saw in Season 7 just how much she struggled with having that power. I very strongly disagree with the idea that she just shrugged off other people's lives.
I don't mean that she shrugs off killing people. I just mean she seems to shrug off:
Angel... too many to mention, multiple murders, torture, abduction, trespassing, destruction of property, rape(?).
Spike... almost the same as Angel, too many to mention.
Faith killing the human assistant of the mayor, was an escaped convict (fugitive), numerous counts of assault in LA and who knows where else, attacking police officers, theft, destruction of property, and illegal possession of weapons.
Willow for one murder, attempted murders, breaking out two prisoners, illegal possession of a fire arm, harboring of a fugitive (Faith), destruction of property, assaulting Police officers.
Xander for theft from an army depot and disturbing holy Indian grounds, illegal possession of military arms.
Giles for murder of Ben and abduction of a drugged minor (Buffy), and those past things he did as "Ripper".
Andrew, theft of a diamond. Accomplice of Warren's girlfriend's abduction, attempted rape, and death, framing Buffy.
Anya, illegal alien (Denmark, Norway, or Swedish citizenship), does not have a working permit to work for The Magic Box, since she's hasn't and can't file for citizenship or even permanent residence. None of her murders, indirect or not, are covered by human law because she was a Vengeance demon.
LOL! I sound like a freaking lawyer from Wolfram & Hart.
But yeah, if she did the "right" thing on our standards, she would have led them all to jail. But she decided not to. The world was a better place because of her decisions.