If a mathematician would throw away his mathematical understandings in order to prove more results, I worry for his soul. A theorem in ZFC-but-not-ZF is much less interesting than a theorem in ZF, except as a topic in model theory.
Mathematical understanding quite often contradicts intuition. Part of becoming a mathematician is learning when to trust your intuition, and when you start your mathematical career your intuition is pretty unreliable. The thing about AoC and measure theory is that it's notorious for challenging your intuition after you think you've built up a lot of good intuition.
Also, from my experience the vast majority of mathematicians don't care whether their theorems are ZF-compliant or need ZFC, unless it's marketed to logicians or specifically deals with AoC in a certain field.
