“Buddhism, on the other hand, or so it is frequently claimed, denies that the individual self exists, thus that it is illusory.”
“Buddhism” and “or so it is frequently claimed” are two oversimplifications (but not your oversimplification) that do not help elucidate the differences between “Hinduism” (which is another oversimplification) and “Buddhism.” “Buddhism” is a European creation that tries to encapsulate a still-living and active field of philosophical inquiry into a sarcophagi that we can all look at as if it is a relic from the fixed past.
Yes, “Buddhism” is philosophy. And most of that philosophy are later additions/refinements/reformulations of various subjects. “Buddhism,” however, does not sufficiently encompass the way of life of Buddhists (difference between Theory and Practice)
Buddhists, on the other hand, are something altogether different, because they are individuals who have been taught from various sub-schools of thought in the various schools of Buddhism, and their takeaways may or may not be representative of the actual meaning of what they were taught. And, of course, they may or may not actually practice this way of life. Only in the coarsest sense is Buddhism a religion. It doesn’t have dogma (philosophical inquiry and self-experience are implicit in this way of life), and it doesn’t have God/god/gods. The Buddha was an historical figure, and one who became enlightened, which he taught about.
What I am saying here is that Buddhist philosophical schools are still in a process of development, and may have strayed far from what the Buddha may have actually taught (Who knows? None of it was written down at the time). Just ask the Theravādin buddhists. They will tell you that everyone else is wrong about what the Buddha taught.
So, what do you want to focus on here? Annata (non-self), Anicca (impermanence), and Dukkha (suffering)? I ask because they are inseparable, according to what is known about the Buddha’s actual teachings. Are you focusing on what the Buddha actually taught? Or some stage of development of Buddhist philosophy?
“The doctrine of no-self seems therefore to be not a teaching of the Buddha; it is either a teaching of a later Buddhist school, or a misunderstanding by later Buddhists and other commentators, or a combination of the two.”
There are clearly teachings by the Buddha on Annata and Anicca, so that statement isn’t correct. But when you move over into Buddhism, then there are very different — in the sense of philosophical development — ideas and logical arguments in each school. In fact, the “progressive stages of teachings on Emptiness” is a thing that is taught in some schools (those that feel their position is more refined).
At its very purest sense though, Annata says that there is no permanent self, like a soul, nor even a permanent self-essence, but that there is Buddha-Nature, which is not personal, and not a permanent entity, so not Brahma or Atman. But Buddhism isn’t atheistic because it doesn’t exclude the idea of God — look at Tibetan Buddhism, in which there are many, many deities — it just asserts that this ‘God’ or these ‘gods’ can’t have a permanent self, nor even a permanent self-essence, so they don’t really fit the ‘God’ role.
Impermanence is very much akin to the Ship of Theseus question, and is based upon every moment of our experiential lives and every scientific bit of knowledge — there is nothing that is permanent.
No self though, is unique to Buddhist philosophy.
And what these philosophical inquiries are focused on is the direct meditative experiences that anyone can have for themselves if they actually do any of the manifold, and well-developed, practices that each school teaches.
So, the answer to your two questions under consideration is: 1) Yes, and 2) Yes.
PS, you have on occasion stated that I am Buddhist, which is wrong. No Buddhist I know calls me a Buddhist. I just really appreciate the unique philosophical understandings and meditative practices (which I find to be very coherent in many cases, with my own), and the holding of compassion as the highest ideal, and, blah blah blah that one finds in the various schools of thought that people with a different way of life have jammed together and labeled “Buddhism” mostly so that they can gloss over it quickly and move on to something they find more interesting 😊
