> the great benefit of the y-combinator is that it permits indefinite iteration in a system that seems at first glance to not support iteration at all

This is only the case for untyped systems. If we try to type the Y combinator, we will eventually need some notion of recursion. For example, in Haskell/OCaml we can type it with iso/equi-recursive types. How many truly untyped systems we use in practice?

The Y combinator as a counterexample of termination is a great insight though!