Set theory as a toolkit is mostly geared towards questions of consistency/existence, size or "order" of objects, and raw mathematical explanatory power of theories or claims. It traditionally arose to explore ideas of infinity precisely, then grew to meet the needs of a mathematical community grappling with antinomies (e.g. Russell's paradox) and heated foundational divides (classicism vs intuitionism, etc). It still meets those needs well -- topics like large cardinals which chart the territory dangerously beyond the boundaries of (assumed) mathematical consistency are plied with set theory, and questions of consistency that arise in other fields tend to end up ported into set-theoretic terms.

I'm not a type theorist, but I imagine the split between set theory and type theory is analogous to the way in which different models of computation appeal to researchers on different sides of the "Theory A/Theory B" split in computer science. Researchers in the "A" group who study the outer limits of what problems are/aren't tractable, how different kinds of computational resource compare in power, etc, tend to use Turing machines and RAM models and circuits, which allow computational resources to be easily measured, bounded, simulated, reduced into things, etc. Researchers in the "B" group who explore programming languages, type systems, program analysis, etc, tend to prefer models like lambda calculus which allow programs to be easily composed and annotated with types and whatnot. I don't think this is just about cultural differences -- if you try to prove time-space trade-offs with the lambda calculus or invent a new type parameterization system with Turing machines then you're going to have a bad time.