because we'd have more time to think about existential risk mitigation while we rebuild society."

A more likely result: the religious crazies will take over, and they either don't think existential risk can exist (because God would prevent them) or they think preventing existential risk would be blasphemy (because God ought be allowed to destroy us). Or they even actively work to make it happen and bring about God's judgmenent.

And then humanity dies, because both denying and embracing existential risk causes it to come nearer.