You fall for a common fallacy here: We do not understand intelligence at all.
No, I don't fall for any such thing. I make that exact point. Read again.
they cannot identify any non-physical phenomenon.
....nor is there any reason -- at all -- to postulate the existence of such a thing until we have ruled out the physical; and since we know the physical is real, and that the brain is physical, and that its workings are, thus far, entirely known to be physical, there is every reason to expect that when we do understand what it does, we'll find the answer the same place humankind has found EVERY other answer ever in our history: in the physical world. Your position is 100% identical with "I don't understand it, so it must be some force I cannot see." Before that answer can be taken seriously, we must reach "We have examined, and completely understand, all physicality and have not found thought or intelligence in any of its workings -- there MUST be something invisible. We're not even close to that point; so the presumption that there is an invisible, unknown force is wholly premature.
But the brain is an analog computer and every quantum effect has an influence on its workings.
Objective data argues otherwise. I can sit in the presence of an enormously strong RF and/or magnetic field that permeates my body, and I can think just fine. Were anything as touchy as quantum effects acting as active mediators in my thought, I would be reduced to a vegetable, or perhaps hallucinate wildly. People work in such environments all day long, every day, and their brains just keep on doing exactly what they usually do. Likewise, physical motion causes all manner of slight stresses to the physical structure of the brain, and yet, it keeps on working. All of this -- and more -- argues for an extremely robust system that is immune to all but the most profound effects. Likewise, the common and eminently predictable effect of a huge range of drugs, injury, surgery upon consciousness and the brain's physicality argues for mundane physicality. Whatever is going on responds in a most typical, physical way. The obvious conclusion -- possibly wrong, but as I said, not the way to bet -- is that it, itself, is physical in the sense that everything else is.
Come on, this is beginners stuff. Do you really claim you do not see that fundamental difference?
Please. Do not mistake my position for an uninformed one. That would be a very large mistake.