You are right; the limited expressive capabilities of first-order alnguage impose the use of axiom schema, like Separation and Replacement in ZF (Zermelo-Fraenkel system) and Induction in (fist-order) Peano arithmetic.
Wherever you need to quantify a predicate (an "attribute"), in first-order logic you must use a schema, with the limitation that while (with standard semantics) attributes in a countable domain are uncountable (i.e. if the domain of individuals has "aleph_0" elements, the possible attributes are as many as all the possible subsets, i.e. have the cardinality of the power-set of the domain), the possible "instantiations" of a schema are countable (we have only countable many formulas in the language).
But, regarding set theory there are more than one axiom system : see in SEP Alternative Axiomatic Set Theories; specifically the NGB system (von Neumann-Godel-Bernays) is finitely axiomatizable:
The resulting theory is a conservative extension of ZFC: it proves all the theorems of ZFC about sets, and it does not prove any theorem about sets which is not provable in ZFC. [...]. two comments about this. First, the mental furniture of set theorists does seem to include proper classes, though usually it is important to them that all talk of proper classes can be explained away (the proper classes are in some sense “virtual”). Second, this theory (especially the version with the strong axiom of Limitation of Size) seems to capture the intuition of Cantor about the Absolute Infinite.