"But from my understanding he demonstrated that it is possible to encode the statement "This statement cannot be proved" using Gödel numbers..."
Not quite. What he did in effect was to encode the statement "This statement cannot be proved by proof system X" where X was a particular set of axioms and rules of deduction, which he could encode using precise arithmetical relationships. People had argued against paradoxes being a genuine problem by saying that English language was ambiguously defined and self-referential, and if you could make it precise enough the paradoxes would be avoided. That was the whole purpose of mathematics and logic. That it could be encoded in arithmetic told us we couldn't keep arithmetic and at the same time get rid of the paradoxes.
The caveat "This statement cannot be proved by proof system X" gives us a loophole to avoid the paradoxical self-contradiction. Any proof system apart from X that proves the statement does not give rise to a paradox. That's how humans can look at the statement and see intuitively that it is true, and not contradict themselves, because humans don't do proofs by the precise formal method Godel encoded.
However, the technique can be applied (informally, at least) to any prover, including humans. The statement "This statement cannot be proved by humans" could in principle be proved true by a non-human AI, but if a human could prove it then they would have proved something that was false, and would therefore be inconsistent, or they couldn't prove it and human proof is thus incomplete.
We can do stranger versions. "This statement cannot be proved on a Tuesday." "This statement cannot be proved by anyone wearing a hat." And so on. There is nothing special about human reasoning with regard to Godel-like statements. Indeed, it is widely known and entirely uncontroversial that informal human reasoning is both fallible and incomplete.
Interestingly, the same principle is the basis of a well-known puzzle - the problem of the unexpected hanging. A judge condemns a prisoner to death by hanging at dawn of some day of the following week, but stipulates that it should be a surprise; that the prisoner should not be able to deduce beforehand that he is to die the next day. This leads to an apparent contradiction - the prisoner cannot be hung on the last day, as he would know the day before, and hence he cannot be hung on the second to last day, and so on. He cannot be hung on any day! And so it comes as a complete surprise to him when he is hung on Tuesday. By setting conditions in the definition that refer to the evaluation process, we can make it possible to define quantities that can only be consistently evaluated in specific circumstances, or by some people and not others.
Of course, humans are not constrained to be consistent in proving or evaluating, which means we can usually figure it out anyway. But that doesn't pose any challenge to the idea behind Godel's theorem :- if our reasoning is consistent then it is incomplete. That we often reason inconsistently is not news - it's why we invented formal axiomatic mathematics in the first place!
