While I completely agree that set theory is not special nor the foundation of mathematics (there is no such thing; foundation of mathematics is a name given to any formal language that can express all or most of mathematics), you are also repeating a common misrepresentation of set theory. While it is true that people often present, say, the natural numbers as some arbitrary encoding in sets, that is not how the natural numbers must be represented in set theory, and set theory can nicely capture the idea of an abstract inductive definition of the natural numbers, as well as the isomorphism between different representation. Many set theories employ Hilbert's epsilon (choice) operator, that allows one to "choose" some set that satisfies a proposition[1]. The question of the members of the set is completely unanswerable, and so can be reasonably said to be nonexistent. You do not work with a particular encoding, and you can't even if you wanted to, because the choice operator does not "reveal" what it is.

It is true that the fact that "there exists" (in a very nonconstructive sense) some unknowable, impenetrable encoding may offend the aesthetical sensibilities of some people, but type theory does not really "resolve" that in any way (and it is certainly not superior, just different, or rather, it is superior in some ways and inferior in others[2]); it simply constructs its axioms at a higher level of abstraction. The axiomatic existence of inductive data types in type theory is just a theorem about well-founded sets in set theory, but those sets are really defined in exactly the same abstract way.

[1]: So the natural numbers can be defined as "epsilon set N, s.t. N is the minimal set s.t. an element zero exists in N and there exists a function succ in the set N -> N, such that for all n in N, succ(n) is in n, and there is no n in N s.t. succ(n) = zero".

[2]: For example, because most type theories are based on the lambda calculus, they are on the precipice of inconsistency due to Curry's paradox, which makes some aspects of working with them extremely unpleasant compared to set theory.