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)
Not sure it’s a tautology. It’s not obvious that a recursively enumerable procedure exists for arbitrary formal systems that will eventually reach all theorems derivable via the axioms and transformation rules. For example, if you perform depth-first traversal, you will not reach all theorems.
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”)?
It's "obvious" in the sense that it's a trivial corollary of the completeness theorem (so it wouldn't be true for second order logic, for example).
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?
