Assume there is either a proof for, against or for independence for every statement within the Peano axiom system.

For every program P you could create a statement S(P)="does this program ever halt?". Now you could solve the halting problem: For given P, iterate over every proof and check whether it is a proof for either S(P), not S(P) or S(P) is independent. Once the machine finds a proof for not S(P) or S(P) is independent, it stops with false. If it proves S(P), it stops with true. It will necessarily stop according to your original assumption.

This works even if you would look for an arbitrarily long chain of "it is not provable that not provable that not provable .... that not provable that S", since every finite proof has to be checked for only a finite number of chains. (Since proof must be longer than the statement)

That is not literally what the original quote says, but an intelligence that could, for example, 'learn' our next century of discoveries in mathematics and theoretical physics in an afternoon seems to me to justify the weaker position that there are possible intelligences that would regard every problem we have yet solved or shown to be unsolvable as obvious and not hard.

That is plausible. But as you mentioned, the original quote, catchy though, was different and unfortunately (unlike most philosophical statements) formally wrong.