Systems of logic and mathematics are designed to validate pre-existing proofs.
“x is a triangle” entails “x has three sides.”
This is known independently of any system of logic or mathematics.
Consequently, systems of logic and mathematics are designed to recognize the existence of this entailment-relation.
So it isn’t so much that such systems generate proofs as it is that they identify the principles of which pre-existing proofs are instances.
Of course, such systems, once in existence, give rise to previously non-existent proofs about methods of proof.
But apart from such meta-proofs, systems of logic and mathematics are there to recognize, and render intelligible, independently existing proof-relations.