Pre-print service like arXiv but with private option, so I can correct my mistakes without publicly advertising my mistakes?
Basically, if I understand you correctly, you want proof of priority without publishing.
Well, this problem is well known from middle-ages and renaissance where people wanted to be able to prove they have the earliest solution without revealing what it is (so that if someone finds it, they can prove they were there first).
They often used anagrams, today you probably would use some private/public key combination for this (I am not a crypto expert, you would have to read up on this).
However, to be honest, this will look weird. The current position in science is: if you have done it, publish it and expose it to criticism, and you may gain priority. If you wish to monetize it, you can create a patent that is public, but protects your rights (in theory), or keep it a trade secret and make money out of it.
As for a website that supports timestamps for unrevealed papers, I am not aware of any, but of course if there were, you need to make sure you trust the maintainers. Furthermore, if you had one and if somebody publishes a completed paper somewhere else before your paper goes live, this will not help you. The fact that you had worked provably, say, for multiple years on the topic will at best prevent plagiarism charges when you end up publishing. But this, if it occurs, could be proven otherwise, e.g. lab diary, emails, etc. It will not make you the &quot;first to publish&quot;. Don't forget, there are quite a few cases where people who had publications were superseded by later, but more visible publications. So, even open publications may be vulnerable to &quot;post-scooping&quot;.
So, your question needs to be what one would need such a service for? I therefore doubt that people would set up such a system with obviously limited utility. Of course, this is not a statement that there isn't.
I understand that this does not precisely answer your question, but I hope I was able to put things in context and why your question may not have a satisfactory answer.