Arguments from physics may not help. Here is Bryce DeWitt reviewing Stephen Hawking and G.F.R. Ellis using the axiom of choice in 1973:

The book also contains one failure to distinguish between mathematics and physics that is actually serious. This is in the proof of the main theorem of chapter 7, that given a set of Cauchy data on a smooth spacelike hypersurface there exists a unique maximal development therefrom of Einstein’s empty-space equations. The proof, essentially due to Choquet-Bruhat and Geroch, makes use of the axiom of choice, in the guise of Zorn’s lemma. Now mathematicians may use this axiom if they wish, but it has no place in physics. Physicists are already stretching things, from an operational standpoint, in using the axiom of infinity.

It is not a question here of resurrecting an old and out-of-date mathematical controversy. The simple fact is that the axiom of choice never is really needed except when dealing with sets and relations in non-constructive ways. Many remarkable and beautiful theorems can be proved only with its aid. But its irrelevance to physics should be evident from the fact that its denial, as Paul Cohen has shown us, is equally consistent with the other axioms of set theory. And these other axioms suffice for the constructions of the real numbers, Hilbert spaces, C* algebras, and pseudo-Riemannian manifolds–that is, of all the paraphernalia of theoretical physics.

In “proving” the global Cauchy development theorem with the aid of Zorn’s lemma what one is actually doing is assuming that a “choice function” exists for every set of developments extending a given Cauchy development. This, of course, is begging the question. The physicist’s job is not done until he can show, by an explicit algorithm or construction, how one could in principle always select a member from every such set of developments. Failing that he has proved nothing.

Some physicists want to use the axiom of choice, but some physicists don't.