I have a PhD in physics and studied graduate level mathematics. Naive set theory is sufficient for basically everything except a few instances, where you have to bite the bullet and invoke the axiom of choice. But this happens rarely and typically is only relevant for instances where there are more constructive but less general approaches.

For the most part sets are used as a notational device, you almost never use any axioms about sets explicitly. If you asked most professional mathematicians to give you the ZFC axioms (or whatever) they would need to look them up.