- He does dumb stuff like in S3 Amends; all characters do dumb stuff but Amends tends to leave me cold; the guy survives 100 years of Hell, keeps his sanity, still remembers Buffy and after a Christmas night of The First he's blubbering 'I'm weak and I need to die', pfft, dude show some self-awareness!
I get where you are coming from in that he has survived 100 years of hell and kept his sanity (although it did take him some time to get back to sanity) so he has shown, as Giles put it, "extraordinary... will and character" (which incidentally is why the episode "Birthday" is such unimaginable garbage - but I digress). I think what "Amends" shows us is that while Angel has an uncanny ability to
survive, what he is grappling with is whether or not he
should exist. He is having an existential crisis regarding his return from hell. Why is he here? Why did he come back? Why does he exist? The only reason Angel pulled himself from the rubble in the first place is that the Powers told him that his destiny was to help Buffy. And then he ends up losing his soul, killing her teacher, and causing her pain and suffering. Add to this that TF turns that very mission on its head by saying "You were born to hurt her." Angel decides 'well if that's true then I'll die.' And TF is smart. It prays on his fear. That he could turn again, that he could once again become a monster. And his desire for Buffy is real. He is, after all, still in love with her. Still desires her in a very human way. So it is very reasonable that Angel would think, that if there is even a possibility that he was brought back just to destroy her or hinder her he is NOT going to let that happen.
That is why the magical snow is so important. Buffy doesn't convince him to live. It can't just be about her anymore. He has to believe that there is a greater Power that wants him around. That there is more to this than the dichotomy of will I be good for Buffy or bad for Buffy. That his purpose isn't just tied to her. This is the birth of his greater mission statement. Maybe there is another reason he is spared. And, of course, Buffy's words resonate with him, after all, it forms the basis of what he stands for: "Fighting is strong, it's painful and it's every day." is reiterated in s4 AtS "Nothing in the world is the way it ought to be. - It's harsh, and cruel. - But that's why there's us. Champions."
Anyway, I rambled a bit, but I think it's very in character for Angel (especially at this time in his unlife) to go straight for "well then I'll die" when he thinks he might be a danger to Buffy. Because if it comes down to her life or his he doesn't give much value, at this time, for his own unlife (and in fact, he will almost always choose to protect a human life over his own - see e.g. The Trial - and yes I say almost always because he does do some questionable sh*t during his beige!Angel days). But in Amends, he is not willing to take a risk with her life or the lives of those she loves if he thinks he could hurt them again. So that's kinda my take on the whole shebang.