[occamrazor]

Very good point. To add on this, even category theory is often taught in the language of sets and classes (“A category is a class of objects with a set Hom(X,Y) of morphisms for each pair of objects, etc.”)

It is possible to use categories as basic building blocks instead if sets but, in my anecdotal experience, this us not what the majority of graduate programs in Mathematics do.

It will be interesting to see whether this will change in the next 20 yea.

[spekcular]

They don't do it for good reason: set theory is basically a strictly better foundation for mathematics, and you have to ape all the set theoretic constructions when doing various things anyway (e.g. constructing the real numbers), so it doesn't buy you anything.

This issue has been litigated extensively, and in my view successfully, by Harvey Friedman on the Foundations of Mathematics mailing list, if you want to check its archives.