Mechanization of Mathematics
You're missing what resolution refutation works for -- namely, just propositional logic.

For example, try using that algorithm on "There exist integers x, y, z such that ​x3 + y3 + z3 ​= ​33." 

There are sets of axioms for which the set of theorems is decidable -- the real closed field axioms are the most important example of that, and they in fact also allow quantifier elimination.

There are sets of axioms for which the set of theorems is not decidable -- by undecidability of Diophantine equations, all Σ1-sound extensions of Robinson arithmetic are examples of that.