You distract her while I kill him is about as unambiguous as it can get - IMO.
We are not talking about the method here, but rather about strategic goal. Strategic goal for Giles is not killing Spike because he personally does not like him. Giles honestly thinks that Spike is a danger - on the same level that Ben was. And he did not flinch at killing Ben.
Of course they included undermining Buffy - that was the entire purpose. Buffy is too enmeshed and refuses to see the risk. You distract - I kill, and we both ignore revenge as my main motive.
I disagree here. Buffy's life's purpose is not protecting Spike. As a Slayer her duty is to the world, not to her former beau who tried to rape her and then went out and got better. She was rational enough about it ( ie not mixing personal feelings with her duty) in S2, and she was right. The fact that in S7 she goes all irrational when the world is hanging on the balance might be a mix of post-traumatic stress disorder inter-meshed with Stockholm syndrome, or it might be the work of the First on her, for all they know. There is not to say that the First is not affecting Buffy's judgement - by making her obsessively protective of the guy with the trigger in his head. The First is quite opportunistic entity ('But this will do' - to quote from S2.) It tries to break everybody's barriers, and there is no say that Buffy is not affected as much as Spike, Wood , or any of the Potentials. In fact , it is reasonable to assume after 'Amends' and 'Conversations with dead people' that at the right circumstances anybody can be breached and bent to unconsciously work for the First. Buffy as much as anybody else.
And, again, if Wood is not an idiot - and he isn't show to be one - he is quite aware that the First wants Spike dead and wants him to do it.
Now, this is absolutely unsupportable point of view. Wood is affected by visions of his dead mother, but it only means that his best judgement is compromised and he is not quite his rational self. The whole point of the First being able to make a healthy young girl kill herself from fear is that it can take over somebody's mind and make them irrational. (And no, Buffy, had no right to say what she said about Chloe.)
Why then is it not biased to give Spike the same chance?
Once again, Giles is on board with eliminating Spike at this point not because he despises Spike as a person ( although yes, Giles generally thinks Spike is a bad joke , judging from his reaction in S6), but because at that particular point in the story
Spike is a real danger - soul or not. Every vampire has to fight their demon, but on top of it Spike's being proven to be a walking talking bomb with a trigger set to kill at some point. That's the only reason Giles is going after him - same way he goes after Ben. As for Robin, he as, I tried to explain already, is himself compromised and honestly believes Buffy needs to be protected from Spike even against her will - because she is a Slayer and Spike has history with Slayers.
If it were Willow or one of the potentials would killing them be an appropriate response? I can see keeping Spike restrained or monitored, but killing him is about as obvious bias as I can imagine.
Neither Willow , nor one of the potentials were repeat offenders at that point. And yes, Giles never felt particular affection towards Spike the way he felt about Willow. And yet the fact that Giles places no particular value on Spike's life 'at that point' because he does not see Spike as a potential champion of good -
does not make Giles any more biased or OOC than his killing of Ben. And honestly, nobody is supposed to see Spike as a potential champion of good at this point. That's the whole plot of the season - would he turn evil again, or would he not? That's the suspense. Blaming Giles for not being as blind as Buffy is preposterous. Blaming Robin for hating his mother's killer and not trusting him - while being goaded by the First on top of that - is unfair.
Does Buffy have any agency here or is she just easily manipulated?
Both. She has and agency, and is manipulated. It is a plot device.
I do think the fact that after their hideous relationship he went a chose to get a soul, and returned to help her in any way he could is validating to a white hat. And a mattress in the basement with chains hardly seems like doling out favors. What made situations suspect is Buffy referencing her former relationship with Spike, and even that was pretty minor.
Spike gets a soul by his own choice - but not to help Buffy, or to be the champion of good. The whole season is build around him 'becoming' champion of good 'after' his soul kicks in and he realizes that there is more to having a moral compass than 'to be hers'. That's why he chooses the sacrifice in the end - because with the soul comes consciousness and responsibility. But by no means anybody is obliged to forgive him, believe in him, and risk the lives of their loved ones for him. Buffy does that at her own risk and it is she who is biased - but luckily for her she is the protagonist and she always gets it right in the end. The season is build around her 'grooming' Spike with her forgiveness to became her sacrifice, and it is all well and good, but the whole point of the plot was to keep it in suspense till the very end ( that's why the First takes Buffy's shape in the end and goads Spike into a fit of jealousy). To assume that Robin and Giles were somehow entitled of more knowledge than the audience is strange.