I entirely agree with most of the first portion of your analysis. It's evidently true that we can't see things like inferences, or abstractions or anything of the like when we see neurophysiological images of brains. That's completely true, and shouldn't be controversial. Where I do take issue is when you say:
"So, I'm arguing those capacities of interpretation and reasoning are internal to the operations of thought. They are not physical in any sense".
There's no doubt a massive explanatory gap exists between the data of science and the experience we have of it. But why isn't the process of the operations of thought "physical"? All it means is that physics, as we currently understand it, is radically incomplete and furthermore, will likely never be completed. I can't understand the idea that in between our brains doing something with data, and we experiencing that something as experiential phenomena, that there is something "non-physical" occurring. That would imply, or could be taken to imply, that something non-natural is taking place in between what my brain does, and what I experience.
I think everything is natural, but we have substantive gaps in understanding, given the creatures that we are. I mean, I could say, something "nous-like" is happening with the operations of my thoughts, but I don't understand how to interpret that, other than taking it as meaning something in additional to the physical is happening when I interpret data. Why introduce something in addition to what we already know is radically incomplete on its own terms, namely the physical?
As for Chomsky. Here I can speak with some confidence. You are correct about the "atheism" bit, I got to meet him too, he's really awesome.
:) You are correct that he doesn't make metaphysical commitments outside of his "methodological naturalism", there is only one world, which we try to understand, theoretically, meaning scientifically, naturalistically, as we do with anything else we can more or less study in a systematic manner.
On the other hand, if you take a look at Chomsky and his Critics, where in his "replies" section he talks about Strawson's argument of Real Materialistic Monism, as presented in Strawson's essay
Real Materialism and Chomsky says that "RMM does "ontologize" the methodological stand, in a way that seems to me to be quite reasonable..." (2003, p.268).
He also states in
What Kind of Creatures Are We? that "Galen Strawson develops the first option in an important series of publications. Unlike many others, he does give a definition of "physical," so that it is possible to formulate a physical-nonphysical problem. The physical is "any sort of existent [that is] spatio-temporally (or at least temporally) located)." The physical includes "experiential events"(more generally mental events) and permits formulation of the question of how experiential phenomena can be physical phenomena-a "mind-body problem," in a post-Newtonian version." (p.120)
But Strawson then argues that experience is physical, the problem is the non-experiential aspects of matter or stuff that has no experience (ordinary objects, the universe, etc.), that's the real mystery for him, until he went down the panpsychist direction.
But you're correct, Chomsky prefers to speak of "methodological naturalism" and not "physicalism".
Nevertheless, whether we use the word "physical" or "world", Chomsky would certainly say that there aren't any metaphysical distinctions, they made sense when Descartes proposed them, but that view collapsed with Newton