Category theory gives a structural framework for discussing these things. The various categories live side by side and can be related with functors. This allows a broader view and makes it easier perhaps, to understand that there isn’t a right answer to “what is true” about sets in the absolute.
For sure there would have to be a meta theory of some sort. But, I think that many classical mathematicians would be happy with that meta theory to be weaker, rather than stronger. I don't think there is a need for that theory to have AC. After all, thanks to its independence of from the other axioms, AC does not add logical strength to a classical theory.
(For die-hard constructivists, such as myself, the story is of course different. But that story is for a another time. I am presenting the classical view here.)
But then you would think there is a right answer to “what is true” about categories, and you would face AC again.