The claim that it must not exist [1] despite that overwhelming experience, is the extraordinary claim. I don't know what consciousness is or does, and perhaps it doesn't actually exist, but the belief that it must not exist because if it did exist that would make reductive statements about the brain a little harder, is a religious belief. It's "I can't explain this and it's inconvenient to what I want to believe, so it must not exist."
[1] Or de facto not exist, as in the hand-wavey claim that consciousness is some kind of byproduct of any sufficiently complicated system that arises from a totally unclear mechanism to no purpose or result.
I think the argument you're making is much more reasonably applied in the opposite direction. The fact that we don't know exactly how consciousness works is no reason to jump to conclusions like "oh, it must be quantum-mechanical and/or incomputable", just like the fact that we don't know exactly how life evolved is no reason to say "oh, God did it."
I suspect the "quantum brain" movement is in part so popular because it tricks people into thinking its claims are not extraordinary. That's why I speculated that its insidiousness comes from the fact it's a New Age pseudoscience which caters to people's pre-existing sense of superiority over the rest of the cosmos. I contend that quantum spirituality holds little appeal for materialists.