Forgive me if this question is quite naïve; I've studied axiomatic set theory in the context of ZF, but my knowledge of NF(U) goes little beyond its axioms, what it means for a formula to be stratified, and stuff that I've read on websites here and there.
What makes NF appeal to me (more so than ZF) is that it uses fewer axioms and it resolves Russell's paradox in a way which better matches my intuition of how I think a 'set' should behave; certainly it's more intuitive than Foundation+Separation+Replacement+... in ZF. The downside of NF is that it proves $\neg$AC, which is a shame. But by adding urelements, which I can almost force myself to accept, we get NFU, which is:
- Consistent with the Axiom of Choice;
- Consistent with the Axiom of Infinity;
- More intuitive than ZF;
And according to this page, NFU can "safely be extended as far as you think ZFC can be extended".
Now ZF(C) has its advantages, constructing new sets from old ones and whatnot, but it still hasn't been proved to be consistent. Wikipedia  says: "A common argument against the use of NFU as a foundation for mathematics is that our reasons for relying on it have to do with our intuition that ZFC is correct." $-$ is that all there is to it?
My question is:
Why is ZF(C) the paradigm under which 'mathematics' is done, rather than NFU(+Inf)(+AC), which we know to be consistent?