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.
Carl Sagan mentions this in his writings and on the Cosmos TV show.
For e.g., suppose you make a claim "strawberry tastes sweet", and I have never eaten strawberry before, I'll say "can you prove it?". You'll say "Sure, here just take a bite, and you'll know for yourself". And I respond "Nah, I don't want to do that. Can you still prove it? The burden of proof is on you".
No, the burden of proof doesn't always rest on the person making the claim. Once he outlines the method to verify it, his job is done. The burden of verifying the proof now rests on the skeptic.
There is a way to remember your past lives, if you do a certain penance for 12 years - not impossible, not complex, very simple, but very hard - do not utter a lie for 12 years consecutively. There are no loopholes - if you are born dumb, or lose the ability to speak, that doesn't count. you must be able and willing to speak, but still not utter a lie.
Do it for 12 years without a single break continuously, and on year 13 day 1, you can remember your previous births. Some have done it (hence the claim), but not many in the western 'modern scientific' world care to verify it, and just pooh-pooh it. Of course, it's not a loss for those who made the claim. Whoever wants it can go after it.
Now lets consider a proof for God's existence. Do most religions claim to have such proof in hard terms? I don't think so. The say it's a belief, a faith.
Not saying that the two not have many things in common, but I also see a strong contrast.