The TMs you hear about in your undergrad studies are not even good models for the webserver that served you this comment. Church-Turing is about computing functions over the natural numbers. You can (and usually do) have a Turing complete model of computation in which you can't program Tetris.
Perhaps more relevantly, I think there's a more basic questionable assumption - the idea that Church-Turing is the best computational basis for AI.
I strongly suspect this is not true - because even after all this time, our best abstract models of computation are still extremely brittle and fragile.
In a hand-wavey sense, as soon as you start allowing for arbitrary inputs and context awareness, you're no longer dealing with a simple Turing architecture but with something else entirely.