I agree, the axiom of choice disincentives us from striving for more elegant solutions.
That said, the axiom of choice is always available in Gödel's sandbox of "constructible sets", and by "Shoenfield absoluteness", some results can escape the sandbox.
For instance, the result that every vector space has a basis is equivalent to the axiom of choice. We have it in Gödel's sandbox and we don't have it outside, if we prefer to not use the axiom of choice in our meta theory. But every purely number-theoretic consequence of that result flows from the sandbox to the ambient universe.
In this sense, the axiom of choice can be regarded as a useful fiction. Not necessarily true in a literal sense, but true enough for many purposes.