(In support of your point about QM:)

To contrast with an example of where quantum mechanics is relevant at the level of biology--this is one I'm familiar with:

https://www.sciencenews.org/article/quantum-fragility-may-he...

Unfortunately I'm not finding anything related to anesthesia except for hand-wavy pieces about "quantum consciousness" (anyone, please do correct me with a link or two if I'm wrong). I blame Sir Roger Penrose, if only because him talking speculatively about it (even in a sophisticated, informed way) seems to give so many others leeway to speak far more casually about the same topic, with far less coherence. This is why we can't have our cake and eat it too I guess