Certainly, ZFC is not a pathological constructed example. When a mathematician looks for a foundational set of axioms, ZFC is THE standard choice. It is, and has been, for the last ~90 years at least.

My point was that foundational mathematics is rarely touched upon by a lot of normal pure mathematics (say number theory, field extension, graph theory).

Interestingly, I believe a lot of people actually dislike the axiom of choice. They find it to be way to 'complex' when compared to the other axioms. It is a lot like euler's 5-th axiom (2 straight lines with the same direction are equidistant everywhere). Interestingly in that case, removing the axiom led to spherical and hyperbolic geometry.