This sentiment seems fairly common in modern mathematics, but it was lacking in Naïve Set Theory. Axioms were presented as facts, not tools. There was little exploration of each axiom, what it cost and what it bought, and what alternate forms are available

Actually that's a quite widespread attitude in the set theory community, not only in that book. Just consider that Hamkins' proposal of a multiverse (that is, a plurality of models for the axioms of set theory), is considered controversial. Maybe influential to this state of affairs was a more platonist approach of the founders, who regarded ZF(C) as a way to describe the intuititve concept of set instead of just another formal tool.