Humans have finite brain size and thus limited state (tape).
Turing machine assumes infinite tape. Strictly speaking Humans are not computationally even Turing machine equivalents. Humans have limited memory. Single human can't solve the same class of problems that Turing machine can.
Any problem that any human in history has ever solved could have been solved with finite state machine of sufficient size.
Theoretically, you are right. But does strict Turing completeness matter, in practice? I don't have the required knowledge to counter-argument your last point.
This is the context that was set up by user yters
> E.g. how could we possible write functioning code so consistently if we are limited by the halting problem for Turing machines?
You can't use known limits of Turing machines to argue that humans can do better because we can't.