What you keep missing is that the issue here is not the truth of the Godel sentence, it is our knowing that the Godel sentence is true! That's a totally different animal, because our knowing it within the axiomatic system that is arithmetic lets us use that knowledge in proofs in that axiomatic system, including a proof of the Godel sentence. That's what renders it false, and the axiomatic system inconsistent, and that's why it is not a theorem of incompleteness. I'm just repeating myself, but maybe at some point it will click.