Ah, I think I see where the breakdown is. If you don't mind, I'd like to confirm my understanding your position?
Am I right in saying you think that any supernatural element is inherently dogmatic (used here to mean "anything you're told is true without scientific proof", rather than "a set of principles laid down by an authority as incontrovertibly true"), and that philosophical and ethical teachings of religions are tied intrinsically to the supernatural such that they can't really be extricated, then? And, relatedly, that there is a meaninful (and potentially absolute) difference between a religion and a philosophical school, such as that something will only really qualify as one or the other?
I don't think that is quite where the meaningful distinction is.... ehh. Let me try to explain.
Thing is... Certainly, you can have dogmatism entirely without religion. Indeed, it has been observed how orthodox Marxism often behaved like religion that way, with references to established dogma and, if you will, a sort of holy books and a prophet. That is why I said I don't know how much Charvaka was a creed of faith and how much a philosophical school - because certainly, even entirely non-religious, materialist philosophies can become dogmatic and, in a way, creeds of faiths in themselves. But on the other hand, you also have philosophical schools which are
not that. Or even philosophical schools where one part takes it as a creed of faith and the other part not.
So really, what is actually opposed to the scientific principle is dogmatism, which isn't reserved to religion alone. However, I do think that religion is, necessarily, a subset of dogmatism, because even in its more open and liberal variations, the a priori requirement for something to be a religion, a faith, a belief, is indeed believing in something. And this something is an article of faith, rather than the result of a deduction, much less an empirically tested hypothesis. Now, of course, none of us empirically tests every little thing in our daily lives, but religion raises that to its core concept -
faith.
And yes, I absolutely do think the ethical teachings of a religion are intrinsically tied to its supernatural content. That is their metaehical legitimization: Because that is what God said, or because that is the way to Nirvana, etc. Now you may often come to the same
conclusions as other ethical systems, but that is true in ethics in general: Murder is bad both in deontology (where it is intrinsically bad) or in utilitarianism (where it produces negative utility), and yet those are two different systems. And with religion, well, murder is bad because God said so, or because bad karma, etc. So religious ethics as a system absolutely do require the supernatural, and you can see that in practice with laws and commandments which are, well, much more specific than "Thou shalt not murder". In the end, laws against not plowing the field with an ox and a horse, or very specific fast dates, or the mandate to maintain very ancient and obscure temple rituals can really only be legitimized by the supernatural, i.e. (in most cases) "God/gods/the spirits/the saints said so".