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it would be impossible for us to understand Gödel's incompleteness theorem

I will cite directly from Jimbo's bag of stuff:

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Part 1) Axiomatic system cannot actually prove all theorems in natural number theory. Note that in real mathematics, we are not encountering many of those for some reason and the one Gödel provided had a self-referential relationship, and it can evidently be proved OUTSIDE of the given axiomatic system.

No minds are involved unless minds are theorem provers working on number theory described by first-order-logic only. Which ain't the case.

Part 2) The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Consistency of axiomatic systems is not a great concern of the real world. And consistency can be proved outside of the system. See also: Gentzen's consistency proof.

That argument was just Hollywood-level bad and pop science from the get-go. Let it rest.