The debate over the philosophical implications of Godel's Incompleteness Theorems seem extremely misguided to me. They are, in my opinion, a proof that not all is mechanism, but it has nothing to do with humans having special oracle powers; that's stupid.
What Godel's theorems suggest, by telling us that a system cannot prove its own consistency but can only have its consistency proven via some larger system in which one embeds it, is that the world is ontologically not simply made up of analytic truths.
Instead, analytic truths are things that must ultimately be built up by starting from postulates and going from there, and that these postulates, insofar as they create a system that isn't merely arbitrary, in some way aid human/technological action.
And therefore, when we do hit some undecidability in a system, the choice of how to fill that gap has some kind of guidance because there is already something in the real world that system is embedded in and we're just further formalizing with the help of some constraints.
But humans as magical oracles in a vacuum? Of course not, that's like suggesting a pigeon could win a race against an SR-71. On the other hand, there is a way humans could win, and that's if they know the space surrounding the analytic system better.
So it would be as if the bird found some portal that made it win the race against the airplane. More importantly, if something is undecidable, that's not a question of the problem being unsolvable, it just means that it has an undefined answer, this is not about "better".
One worrying about whether there are math problems humans can't solve is a facile issue; that's just a question of whether or not the problem has an answer given the context—adding the context to solve it is just adding the sufficient context, i.e. synthesis.
So no, there's no special human superpower of any kind, it's simply that computation is a different world than ours, or particuarly a subset, something like a 2-dimensional sliver in 3-dimesional space. You don't have special powers in that 2-dimensional world, there's simply different stuff.
Again, none of this is about magical powers, humans in fact suck at computation. It's simply that analytic statements/mechanisms are not the whole world but something we build and integrate into our world, and Godel's theorems mean that the hard work will never end.
This is not simply a matter of abstraction: mathematics/computation is the construction of what most of us would call "rationality" or "objectivity" via the construction of stable systems of entailment, and the formalisms we see are but one expression of this.
What do I mean by an "expression"? Well, there are other ways that these rigorous sytems of entailment exist: think about actual computers, or systems of currency, or just our neural/cultural wiring that enables counting and arithmetic; that too is "formalism".
In other words, analytic truth is form, and form is constructed, but not in any way "imaginary".
And we're so good at this that it now looks like everything was just made this way for us by God, but that simply isn't so, because if that were true we'd be able to hit a bottom. And as we see from the aforementioned theorems, there is no such bottom.