The answer is that, although it may be a mistake to think that all sets are wellfounded, it is not a mistake to think that the concept of wellfounded set is worth axiomatising.
Perhaps surprisingly, given that constructions like this provide complements for all sets, they give a central rôle to the concept of wellfounded set in that the wellfounded sets of the new model can be made to be an isomorphic copy of the wellfounded sets in the structure one starts with.
(Replacement for, say, wellordered sets doesn't come free in the same way) ZF arose as an attempt to axiomatise the theory of wellfounded sets, and this principle is kept constantly in mind by the ZF-istes as they struggle to find new and more informative axioms.
Aczel used this technique to provide models of Forti-Honsell's antifoundation axiom [1983]; Hinnion's idea was to use isomorphism classes of wellfounded pointed extensional relational types to interpret theories of wellfounded sets in NF.
To the extent that mathematical concepts can be implemented in the theory of wellfounded sets (aka ZF) they can also be interpreted in NF – since NF has a theory of wellfounded sets.
In the 1970's Boffa and Coret proved some interesting results about stratified formulæ in theories of wellfounded sets, and work continues.
The axioms of ZF can usefully be thought of as arising from an attempt to axiomatise the theory of wellfounded sets.
For example, one can restrict to universes that are ω-models, β-models (i.e., wellfounded), etc.
As we have seen, implementing mathematical objects (ZF-style) as wellfounded sets and then reasoning about those wellfounded sets) is not a very satisfactory way to proceed, since NF does not have sufficient strength for us to manipulate the implemented objects with the freedom we want.
This project has been so successful that nowadays many people believe that wellfounded sets are the only kind of set there is if it isn't wellfounded it isn't a set but a hyperset perhaps.
