> Do most religions claim to have such proof in hard terms?
Do most people into astrology claim to have scientific proof in hard terms?
I know a lot of people casually (and some pretty seriously) into astrology, none of them claim that. It doesn't stop them from believing it. They just aren't thinking in a scientific mindset at all. They use confirmation bias to prove it to themselves anecdotally, and many also have a deep irrational distrust of science that has really not been thought through.
So I think holding astrology people to scientific standards is just as meaningless as holding religious people to scientific standards. And plenty of religious claims are eminently testable: the earth was created in 7 days, some holy book is the literal word of god and not written or even tampered with by humans, some religion is the one true path and all others are evil, etc. These things admit themselves to clear debunking scientifically and/or rationally.
To be both religious and scientific means retreating your religious beliefs to all but the most abstract and untestable ideas, something like "god is the universe itself" or other such ultra-generalizations that leave no trace of casual claim of interference with material reality. And the vast majority of religious people on earth do not do that.