I guess it is tautological from the definition of "provable". A theorem is provable by definition if there is a finite well-formulated formula that has the theorem as consequence (https://en.wikipedia.org/wiki/Theorem paragraph theorem in logic)
Hilbert’s program was a (failed) attempt to determine, loosely speaking, whether there was a process or procedure that could discover all mathematical truths. Any theorem depends on the formal system you start with, but the deeper implicit question is: where do the axioms come from and can we discover all of them (answer: “unknown” and “no”)?
Hilbert's program failed in no contradiction to what GP wrote because the language of FOL theorems is only recursively enumerable and not decidable. It's obvious that something is true if you've found a proof, but if you haven't found a proof yet, is the theorem wrong or do you simply have to wait a little longer?