My original reframing was: the only way of showing that artificial intelligence can be constructed is by showing that humans cannot compute more than the Turing computable.

I was assuming the word 'compute' to have broader meaning than Turing computable - otherwise that statement is a tautology of course.

I pointed out that Turing computable functions are a (vanishingly) small subset of all possible functions - of which some may be 'computable' outside of Turing machines even if they are not Turing computable.

An example might be the three-body problem, which has no general closed-form solution, meaning there is no equation that always solves it. However our solar system seems to be computing the positions of the planets just fine.

Could it be that human sapience exists largely or wholly in that space beyond Turing computability? (by Church-Turing thesis the same as computable by effective method, as you point out). In which case your AGI project as currently conceived is doomed.