I mean this in the nicest way, but I would never have dreamt to relate Gödel's incompleteness to this topic.

I think perhaps where I would diverge is that we should separate the language from the program. Generally, we say that a programming language is Turing complete.

Gödel's incompleteness is analogous to a paradox, e.g the Liar paradox, that is one useful way to understand the statement. But it seems as if you're saying it's a Contradiction (from where we can derive anything) something like EFQ: https://en.wikipedia.org/wiki/Principle_of_explosion

That does not follow, not all paradoxes are contradictions. For instance, just because Gödel's incompleteness pertains to first-order Logic does not mean that we can derive anything in first-order logic.