What Gödel did prove was that the above list of statements (and infinitely more) must exist.
The special thing about his proof is indeed that it is recursive, i.e. adding another axiom can't fix the problem. But that was only necessary to prove that no formal system can be perfect, ever.
Most undecidable problems can indeed be "fixed" by adding another axiom, but if you go beyond the ones of ZFC it becomes less clear which of the two alternatives is the "right" one...
What Gödel did prove was that the above list of statements (and infinitely more) must exist.
The special thing about his proof is indeed that it is recursive, i.e. adding another axiom can't fix the problem. But that was only necessary to prove that no formal system can be perfect, ever.
Most undecidable problems can indeed be "fixed" by adding another axiom, but if you go beyond the ones of ZFC it becomes less clear which of the two alternatives is the "right" one...