> Doing mathematics doesn't really depend on foundational logic in the same way that, e.g., a web app depends on transistor physics.

Web apps don't depend on transistor physics at all, though. An important consequence of Turing universality is that computer science is not a subfield of electrical engineering.

Also, mathematics underwent very dramatic transformations during the late 1800s and early 1900s alongside the early development of formal logic and set theory. To say that logicians were merely "formalizing the intuitions behind the math that people were already doing" strikes me as misguided at best. The mathematics that rose to prominence in that era was very different from what preceded it, often controversially so.