I think you're taking this point a little too forcefully; this is meant to informally motivate Russell's paradox, in my reading - which is exactly the title of the section you're referencing.

The point here is a little more subtle; category theory doesn't necessarily rely on sets; the definitions of categories that you often see (involving sets of objects and sets of morphisms) is more axiomatically forceful than the more general definition, which uses the notion of classes; category theory can use set theory, but does not depend on it.

The point here is that type theory offers just such another way to design in an avoidance of Russell's paradox.

You might also want to read about e.g. Grothendieck universes - they're quite relevant here.