In a world where it's taken for granted that, say, countable additivity over measurable sets will work work out fine, and is felt to be a different beast than arbitrary use of a separation schema plus a set of all sets, sure. But it took early 20th-century mathematical logicians decades of exploration to get to a point where there was confidence that people doing the former thing could get by mostly "naively", without being in danger of doing the latter thing. (See e.g. Galileo's paradox.)

Modern pure mathematicians benefit from being able to conduct most of their reasoning on the lower footsteps of the von Neumann hierarchy (V_{omega + k} for small k), where they get access to higher-order tools like function spaces and topologies, countability as a fruitful property to impose and work with, and a playground where most of the disparate branches of mathematical research can be brought together without worrying about bindings between underlying frameworks. None of these things require a deep knowledge of the metamath of ZF to work with, but they'd all make many 19th-century mathematicians nervous.

Note also that the most frequently used tools of naive set theory today (e.g. unions/intersections, separations/replacements, powersets, cartesian products/relations/functions, a few notable infinite sets) roughy mirror ZF. That shouldn't seem like an accident. Having working mathematicians trained to think and organize their work in these constructs keeps them around those lower footsteps.