If I said that recovering from cancer does not act according to classical physics, or even according to conventional quantum mechanics, you would consider me a delusional wacko with nothing useful to contribute to the conversation.
Given the weight of evidence showing that anything we can measure about the brain can be explained by physical processes, the onus is on the person claiming that's not how it works to point to some phenomenon that isn't well explained by classical physics.
Generally overconfidence in our present abilities isn't conducive to stimulating new research.
I mean: I give you as much workup on the brain of an unknown individual as you care to have (fMRI, scans, dissection, whatever you want).
Can you detect which languages, if any, the person speaks (or spoke), with any robustness?
Can you detect if the person ever saw an ocelot in person?
Can you detect which sports team, if any, they like or liked to root for, or which instruments, if any, they played?
For most sections of the brain the level-of-detail at which functionality is decoded is still very primitive and coarse-grained. The reason god-of-the-gaps arguments have so much traction is b/c the gaps are still so very large.
Given the weight of evidence showing that anything we can measure about the brain can be explained by physical processes, the onus is on the person claiming that's not how it works to point to some phenomenon that isn't well explained by classical physics.