[aruss]

Category theory isn't a type theory (in the sense that you can have the category of categories, which correspond to different type theories).

It's true that you can get away from Russell's paradox in category theory, but that's because categories are not sets and have different properties (there's less you can say about them because they're more general).

Type theories, particularly homotopy type theory, are becoming popular among some mathematicians, but not because they escape Russell's paradox (I'm not sure that they can say anything more about sets than set theory), but because they are more useful when constructing automated proof systems.

Edit: also, category theory is used a lot in very cutting edge math, like algebraic topology.