I mean a computer science perspective on the Axiom of Choice itself.

While I am naturally much more sympathetic to intuitionistic/constructible mathematics than the general math community is, I won't go so far as to insist it is the only one true math, since I don't think there is such a thing. I just think that from an information theoretic perspective, calling the instantiation of infinite unknowable amounts of information into a proof "aesthetic" is, well, not a term I'd use. I find Axiom of Choice generally gross.