As far as we've been able to determine, almost all macroscopic objects can be behaviorally described in terms of classical physics, and this includes neurons.
Even in rare cases where quantum effects are seen, such as photosynthesis, or radioactive decay, it's extremely localized. The fact that a plant's efficiency of conversion of light to chemical energy appears to use quantum effects doesn't affect the rest of the plant or turn it into some "Schrodringer's cabbage" thought experiment. We can still understand the rest of the plant in classical terms.
Even if it were to be discovered that neurons are sometimes (or even always for that matter) behaving randomly due to quantum effects, that would not mean that we have free will and are not automata. It would just mean we are non-deterministic automata.
There can be macro objects with non-determinism. The cat in the box in the Shrodinger experiment has deterministic neurons but you don't actually know the state of the cat as a whole due to the initial quantum event that set the entirity of the cat on one path or another.
This is where the entire problem with the thought that systems are deterministic or not comes into play. If there's a single photon that can hit a pixel on a sensor (or retina) or not and that ultimately has a cascading effect to make that larger object do something completely different or not then that larger object is non-deterministic as a whole. Yes as others have stated there's ways in which non-deterministic inputs can end up not mattering - the system has the same output. But there's also systems, eg. the cat in the box experiment, where the macro object ends up taking on the non-determinism of a single quantum event.
An AI that can behaive one way or the other based on what it read into the sensor input is absolutely as non-deterministic as that input. Likewise humans are as non-deterministic as their inputs. If a photon can hit your eye today and cause a flash of light that sets you on the path of doing something you otherwise wouldn't have done it doesn't matter if your neurons are essentially a machine or not. You are still non-deterministic. The focus on neurons being deterministic or not kind of misses the bigger picture. We absolutely live in a non-deterministic world and those inputs from this world cause ourselves to ultimately be non-deterministic.
This really gets down to the regularly discussed thought experiment of simulating a human. If you fully simulated a human with non-deterministic inputs everywhere you probably would indeed have a fully deterministic system. But that's just not how the real world environment a human is in behaves.
It's not at all clear how quantum mechanics in general, or a thought experiment like Shrodinger's cat, should be interpreted.
In the lab one can perhaps coax a few atoms near absolute zero into a superimposed state, but not a large room temperature object like a cat, so that particular thought experiment is just that - not an observed experimental outcome.
Is quantum randomness actually real, or just a reflection of an imperfect model of reality ?
In any case, isn't William James really just considering the issue of free will ? If the world, including ourselves, is just an automata evolving according to (say) Schrodinger's equation, then does it really make any difference if it's deterministic or not ? Even at a classical level the future is unpredictable due to chaotic dynamics, and from a subjective (presumably misleading) POV we have free will, regardless of what the reality is.
I don't understand why you're making this argument. It seems to basically be about the semantics of "deterministic" (or not) when composing different systems, which isn't really relevant to the original point.
Just because some macro systems can be non-deterministic doesn't mean they all are. If you have non-deterministic binary inputs to an AND gate, do you really mean to say the AND gate is not deterministic?
The statistical effects that reliably turn individual gas molecule motion into PV=nRT, are just as reliable in biochemistry.
There's an entire field of deterministic mathematics called Chaos Theory, that pretty much encompasses what it says on the tin.
It turns out that -innocent, simple looking- finite state deterministic automata can actually be less predictable than the non-deterministic kind. (and it can get pretty wild! https://www.youtube.com/watch?v=C2vgICfQawE )
Sure - not everything that looks random is random. How about Stephan Wolfram's finite automata "Rule 30" for another example.
But anyways, neurons don't even APPEAR to be acting randomly - they appear to obeying the laws of chemistry, and behaving in meticulously studied and well understood ways.
Even in rare cases where quantum effects are seen, such as photosynthesis, or radioactive decay, it's extremely localized. The fact that a plant's efficiency of conversion of light to chemical energy appears to use quantum effects doesn't affect the rest of the plant or turn it into some "Schrodringer's cabbage" thought experiment. We can still understand the rest of the plant in classical terms.
Even if it were to be discovered that neurons are sometimes (or even always for that matter) behaving randomly due to quantum effects, that would not mean that we have free will and are not automata. It would just mean we are non-deterministic automata.