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I've always read this part of Good Omens as a bit of moral cowardness on the part of the duo. A life against the universe is a very utilitarian way to see things, and if you subscribe to this philosophy, it's not wrong. Both would like it to be no antichrist, but neither are willing to do the deed themselves.
And they both have their reasons to not engage too deeply with moral matter. Aziraphale principal failing is having much more in Heaven than is warranted and believing them (or trying to believe them) to be right. Tautologically. (There's a reason why the sarcasm in "You're an angel, I don't think you can do the wrong thing." doesn't register). So as much as he doubts the righteousness of the Flood, or the Crucifixion, he can fall back on "Well, if Heaven thinks it's okay...". Hence why he's so desperate to have their approval way after it's obvious to everyone that they're not interested in stopping the apocalypse.
Crowley has it a bit easier, since his side is supposed to be wrong, and not liking any of them assholes is perfectly acceptable. So if Hell is peeved at him for being more annoying than murderous, it just means he's succeeding at being annoying. Which is a justification for not being murderous. But not having a convenient excuse for not wanting everyone to die, he's much more ready to throw in the towel.
Because Crowley is on the right side of the argument when it comes to their relationship (for starters, by acknowledging they have one), it's easy to forget that in the bandstand scene, he's arguing for giving up on Earth. I think that's Illustrative of Crowley's main problem: with one notable exception, he's not comitted to anything.
He didn't mean to fall, he just sauntered vaguely downard. He's shocked by the Flood, but he doesn't do anything about it. He didn't invent the Spanish Inquisition (and, in the book, gets drunk for a week when he finds out about it) but he'll take the credit. On his day to day, his demonic activities consist of putting a large number of people in a bad mood, which admittedly is more efficient, but also means that he doesn't have to do or witness any of the really nasty stuff that could result from it. And he wants the Antichrist to be gone, but he doesn't want to have to kill him himself.
I've been working on this post for a while and I am running out of time! So here goes: Today I want to talk about the dark side of Good Omens. This is a long post, sorry about that, but there are so many twists and turns. We all love GO, the romance, the banter, the comedy, the humanity of it. But it does a have a subtle dark side underneath which is usually overlooked. It has to do with the race to stop Armageddon, how their different views brought them to the bandstand conflict, and how Aziraphale ultimately came around at the airbase. It was a rocky road. In the book, when Crowley receives the basket and is on his way to the hospital, he has a thought. We didn't see this in the show so it may or may not have happened there, but it happened in the book. Ultimately Crowley didn't go through with it, but it did occur to him.
So Crowley kept driving and eventually came up with Plan A: Raise the antichrist together behind Hell's back. Surely as a normal, balanced human being the kid wouldn't want to destroy the world, easy and straightforward, right?
Aziraphale's orders are to keep Crowley under observation and so he does. But he also tells Heaven he will "influence the child towards the light". Heaven humors him as long as he keeps doing his job: monitoring Crowley and the antichrist.
However as the scheduled time gets closer, Crowley realizes that the only way to be 100% certain there won't be an Armageddon is to not have an antichrist at all (like 11 years ago). And he brings up Plan B: Kill the child, for the first time.
Aziraphale didn't like the idea of killing Warlock outright if there were other options. So he came up with Plan C: stop the dog. That... was very short-lived.
And here is where things will start to get murky. While both know they need to find the antichrist, they don't really know what to do afterwards.
Crowley probably thinks that whatever they do, they'll do it together, that Aziraphale will be on board with it. Aziraphale on the other hand still thinks Heaven are "the good guys" so he separately comes up with Plan D: Tell Heaven where the antichrist is. If Heaven gets rid of Adam, all will be fine. Heaven knows better and thy are going to win anyway. At this point, Aziraphale is fine with eliminating the antichrist, he just doesn't like the idea of doing the deed himself, let Heaven handle it. So he lies to Crowley about finding Adam.
And he lies again about having found the antichrist when they meet at the bandstand. Aziraphale has made his mind. As they talk, Aziraphale's reasons that if Crowley were to kill the child instead of Heaven, it would be a better solution for everybody. Again, he is fine with eliminating the antichrist, but not with doing it himself, Crowley can do it. Should this be a very short lived Plan E?
Crowley on the other hand, just like at the Crystal Palace, insists that Aziraphale do the deed. Aziraphale would not have it. Either Crowley does it or Heaven does it, but not him.
The book is more clear about how Crowley is afraid of Hell's punishment if he fails. In Crowley's mind, if Aziraphale kills the child it would be ok, Aziraphale gets brownie points in Heaven and Crowley gets punished for maybe only being incompetent.
But if Crowley kills the child, he would be in really BIG trouble, punished as the worst traitor there could be. (this quote comes from a later passage but it is the sentiment that counts)
Both agree that the antichrist needs to go. But Crowley won't kill the child and Aziraphale won't kill the child either. So they come to an impasse.
Aziraphale proceeds with his Plan D anyway and his conversation with the Metatron goes as pear-shaped as could. And this is where he decides to talk to Crowley and reconcile. That didn't work out either.
After a short trip to Heaven, Aziraphale finds himself on Earth and ready to follow Crowley's Plan B. However he still doesn't want to do it himself, so this time he is recruiting humans to do it (Plan F!). Humans are good at killing other humans, they have been doing it for a long time ;)
Said humans are on board with it until they realize that the antichrist is an 11-year old boy and Plan F goes up in smoke.
Crowley is not giving up though and Aziraphale, fully committed now that every other option failed, follows Crowley's lead. Third time is the charm, right? Aziraphale doesn't hesitate anymore and shoots when Crowley urges him to do so.
And, as we know Madame Tracy saves the day and Adam (although for all we know Adam is bulletproof at this point ;) ) So yeah, interwoven with all the fun there are these glimpses of hidden darkness, our demon who keeps pushing Aziraphale to kill the antichrist for lack of a better plan and our angel who doesn't want to taint his hands and keeps looking for other people to do the deed instead. There is certainly a lot more nuance to this, Crowley and Aziraphale are not selfish, detached entities who end human lives willy-nilly, not at all. But in a desperate hour, they will, if that means the survival of the whole world. As Crowley put it back at the Crystal Palace "one life against the universe". He was ready to sacrifice that life from the very beginning. Aziraphale meandered his way to get there, but at the end he got there and shot at the child.
It is certainly very fortunate that they were not very competent at this. It would have been awful killing Warlock and then realizing he wasn't the antichrist after all. And Adam is a nice and smart kid who found his very own human solution to the problem without intentionally shedding any blood. They both deserve to live, and our walking disasters deserve to get their world and their humans without having to pay for it with the life of a child.