If you translate everything to being stated in terms of category theory, it clarifies how it all works and makes it easier for people to go from one area to another. It is a bit like what Grothendieck did with abstract algebra. Imagine if every sub-field of mathematics had their own word for group, and that's something like how category theory fans see the present world.

There's no such thing as a theorem that can only be proven with category theory, because whatever specific thing you're applying category theory to already has all of the properties that would make the theorem true without category theory being involved - if it didn't, you wouldn't be able to treat it as if it was a category. Categories are not the kind of mathematical invention that, for example, ordinary differential equations or matrices are.