There are indeed many people trying to justify this magical thinking by seeking something, anything in the brain that is out of the ordinary. They've been unsuccessful so far.

Penrose comes to mind, he will die on the hill that the brain involves quantum computations somehow, to explain his dualist position of "the soul being the entity responsible for deciding how the quantum states within the brain collapse, hence somehow controlling the body" (I am grossly simplifying). But even if that was the case, if the brain did involve quantum computations, those are still, well, computable. They just involve some amount of randomness, but so what? To continue with grandparent's experiment, you'd have to replace biological neurons with tiny quantum computer neurons instead, but the gist is the same.

You wouldn't even need quantum computer neurons. We can simulate quantum nature on normal circuits, albeit not very efficiently. But for the experiment this wouldn't matter. The only important thing would be that you can measure it, which in turn would allow you to replicate it in some non-human circuit. And if you fundamentally can't measure this aspect for some weird reason, you will once again reach the same conclusion as above.