Structural set theory, which as you say throws out the iterative structure, is more like type theory than set theory (this is pointed out in the nlab page you linked).
Naïve set theory as practised in say a course in discrete mathematics certainly uses the iterative structure to construct say a pair. You will typically find the definition 〈x,y〉 ≔ {{a},{a,b}} instead of defining it through a universal property.
All the axioms of material set theory take this iterative approach, not just foundation. In fact foundation is not very critical (The Axiom of Anti-foundation is just as workable), as you can always just use the class of well-founded sets and work with those.
Naïve set theory as practised in say a course in discrete mathematics certainly uses the iterative structure to construct say a pair. You will typically find the definition 〈x,y〉 ≔ {{a},{a,b}} instead of defining it through a universal property.
All the axioms of material set theory take this iterative approach, not just foundation. In fact foundation is not very critical (The Axiom of Anti-foundation is just as workable), as you can always just use the class of well-founded sets and work with those.