I don’t think it’s a question of ‘belief’. AOC is logically independent of the other ZF axioms: you may find it aesthetically ugly (a viewpoint I’m sympathetic to). But you can’t disbelieve it any more than you can disbelieve Euclid’s parallel postulate.

But I don't have to use it, or I can deprecate it.

Particularly I think those "real" numbers are phony compared to the integers and rationals. (At least the latter have names)

Here's a professor of math in some Australian university who has pursued a philosophy of mathematics while basically rejecting infinities, axiom of choice, and reals.

https://www.youtube.com/watch?v=U75S_ZvnWNk