All good points. I don't know much about his actual higher-level mathematical stuff (I'm pretty bush-league about theory overall, really!) But I've heard that a lot of people in the music theory world say it's awfully close to regular Neo-Riemannian theory.
That said, in the Princeton lecture notes, he does seem to start a given topic on fundamental/"why" questions and then build out, more than any other intro music material that I've seen. A far as I can tell, his own theories are in there, but more as a framework to help explain things, rather than demanding you think that way.
For example, in the first pages, he describes a bunch of well-known fundamentals in music theory/psychoacoustics, but in a very "Tymoszco-esque" way. He lays out five principles common to a huge range of music styles: some combinations of notes are considered more pleasant than others, you don't want to have melodies that jump around too much, etc. (These all go for the Western musical tradition, but also many others.)
Then he says that our scales are essentially "solutions" to how you can chop up the octave into steps and meet all of the five requirements. It turns out there are a relatively small number of ways to do that, we've found just about all of them, and those are pretty much exactly the list of scales that we're using. But then he quickly builds it out to common scales that you can play on a keyboard written out in normal notation, and goes from there.
This is extremely satisfying to my hacker brain, like picking a handful of criteria and doing a search to find all the possible lists of numbers that meet those criteria.
I wouldn't be surprised at all if you can explain music theory from fundamentals just as well using traditional Neo-Riemannian theory, or even something totally different. I just haven't ran into any intro-level materials like that.
(Also: it could even be that his list of requirements isn't really true, but it's hard to see which ones you would want to deny for Western music.)