On July 29, 2022, I registered the domain “***.site”. Moreover, I even add at the registrar, just in case, arbitrary Cloudflare-NS.
However, when I add the domain in the control ("Add a Site’), I get this warning - “***.site is not a registered domain”. Using WHOIS, all registration data of the domain obtained correctly.
Sounds like you pointed the domain to Cloudflare nameservers before you were instructed to. During the setup process, Cloudflare must be able to connect to existing non-Cloudflare nameservers in order to query information that it needs. The registrar you bought the domain from should have nameservers you can use temporarily; check their documentation or look for an option called default / system nameservers or similar.
I honestly find your response slightly inappropriate. Not only did you post a question which has been addressed a gazillion times, @user4358 has actually provided you with the correct answer based on the information you provided (which by far is not complete).
Saying he did not understand your question and his response was useless, is not only rude, but also factually wrong.
Since I had such an inscription “***.site is not a registered domain” in the control panel, when adding the domain, I added the day after registration NS-servers from CF instead of NS-servers of the domain registrar. But nothing has changed.
By the way, I’ve done this before with other domains, and everything was fine.
Cloudflare will give you this message if you change nameservers before you are told to. That’s just how it works.
Change your nameservers back to your registrars, then add the domain and wait for Cloudflare to give you their nameservers.
I meant to do a lookup with a tool like nslookup or dig - Cloudflare’s nameservers will refuse lookups for your domain since your domain is not on Cloudflare.
And you’ve already been told that’s the issue with your site.
There is nothing to discuss or workaround - remove Cloudflare nameservers, use your registrar’s and then add the domain.