It is a fairly interesting fact that First Order Logic (and Propositional Logic) sort of chrystallized out of considerably more powerful and complicated systems (of Frege and of Russell and Whitehead) as systems that were interesting in their own rights.

There are a number of interesting results about FOL, notably by Lindström, that show that FOL is in some - technical - senses optimal given its means and restrictions.

Also, it is an interesting fact that standard set-theory is a first-order theory (but not pure predicate logic, since it is based on the primitive "is an element of").

This also means that in set theory one has a considerable amount of something much like higher order logic, since one has something like the equivalent for any predicate P(x1 .. xn) by defining p = |x1 .. xn : P(x1 .. xn) and using (x1 .. xn)(p)( (x1 .. xn) e p --> .... ) ) where ( (x1 .. xn) e p --> .... ) ) will be a formula about all n-ary predicates. Or briefly, because in 'x e z' one can use 'z' much as if it is a predicate of 'x' and definable by abstraction.

Likewise, having Higher Order Logic one can set up in it equivalents of set theory (without using the set-theoretical primitive 'is element of' or by defining it).