I agree that "set theory" itself is a bit of an overloaded term, as there are in fact many set theories. However, as you're probably well aware, naïve set theo

I agree that "set theory" itself is a bit of an overloaded term, as there are in fact many set theories. However, as you're probably well aware, naïve set theory can quickly lead to disaster because it leads to well-known paradoxes (like Russell's paradox). So if you're going to have foundations for mathematics (and it needs foundations!), it needs to avoid the paradoxes. ZFC does so, and in a way that many people have found satisfying enough.