You wouldn’t even notice if some proof is wrong because it relies on an inconsistency that’s the issue.

You wouldn't notice because there's no realistic chance that any meaningful result in the vast majority of math depends strictly on the particular way in which ZFC is hypothetically inconsistent.

And that’s before you didn’t notice because noone builds anything on axioms but instead uses fragile foundations made of intuition, hand-waving, and mass psychology.

This is a ridiculous attitude. Nobody uses the axioms of ZFC directly because that would be stupid. It's obviously sufficient to know how to do so. There is literally no difference to the vast majority of all math which particular axiomatic formalism you decide to use, because all of those results are trivially translatable between them.