Without category theory, certain generalizations of differential geometry would be virtually impossible to describe in any useful matter. These generalizations are extremely important for applications to physics and elucidate a lot of the fundamental structure of the classical theory. Furthermore, you get a much nicer theory when you consider higher geometric structures.

The power of category theory as foundations is that it lets you efficiently study the structure of a theory and how to modify that structure in principled ways.

Sure, the mathematician who is doing hard analysis probably does not care too much about category theory, but every mathematician being educated today knows at least a little category theory because it is the natural setting for certain key mathematical ideas, like cohomology.

In contrast, Set Theory is powerful but it doesn't organize mathematics the way logic/languagecategory theory does.