I’d say choice isn’t terribly controversial anymore. If we can prove a contradiction using it, we could have done so without it. (That is, ZFC is consistent if and only if ZF is.) I think the prevailing opinion is that this result has resolved any worry we might have over its use.
That being said, there are alternatives. For example, one can instead take the axiom of determinacy. It implies countable choice, but is inconsistent with full choice. ZF+Determinacy is an interesting theory, which differs from ZFC in a number of ways. For example, all sets of reals are measurable.
However, I’d posit that removing choice tautologically gives us a weird paradox: the formal statement of the axiom of choice says that “the product of nonempty sets is nonempty.” If choice fails, then there exists a collection of nonempty sets whose cartesian product is empty.