I think the more fundamental question to ask is why set theorists insist that the axioms of set theory be strictly first-order in nature (*). I claim that you can't really explain the motivations of set theorists until you address this phenomenon.
In light of this near-universal assumption, I think it is fair to say that:
First-order logic is the foundation of set theory (which is in turn the foundation of other things)
The axioms selected for set theory are those which enable as much model theory as possible, without risking inconsistency.
The fact that one can then turn around and use model theory to study set theory itself is a major bonus, but not really a part of the foundations argument. I've heard some quasi-mystical tales about sets existing in some alternate universe out there, but I feel far more comfortable using "it lets me do model theory and hasn't lead led to contradictions" as a justification for the axioms.
(*) Excluding, for example, second-order quantification, the logic $L_{\omega_1,\omega}$ of countable conjunctions, or the "exists uncountably many" quantifier.
