I suppose to some extent this is a matter of taste. I'll just say that, in my experience, people are typically very comfortable with, e.g., logical connectives and the primitive notion of a set of objects from grade school mathematics education. So this framework is "natural" and readily believed.

Further, in ZFC, the only basic notation is that of a set. In something like the calculus of constructions, there are five fundamental notions (if I remember correctly). From the standpoint of ontological parsimony, that's a win for ZFC.

Axiom of separation just says we can make subsets of things - I think this is not so hard to swallow. I'm curious what you find counterintuitive about it.