not quite. most mathematicians pretend they work with ZFC but can't state the axioms. they (we) just work with a naive notion of set and the habit not to write something blatantly inconsistent.
of course, every large formalization effort brings some technical debt.
in the 80s, Pierre Cartier explained in several interviews why Bourbaki didn't make the move to categories some 20 years before while they (Cartan, Eilenberg) had basically if not invented them, then making them prominent: that would have meant rewriting 2000 pages and they felt they had better write down their ideas about Lie groups and Lie algebras.