This is now included in Collect v2021.2 beta 5 and it looks like the release will be out early next week!
Apologies for the radio silence, @Tino_Kreutzer. We're not going to expand the scope of v2021.2 further just now because we're getting at risk of never shipping! Thank you for the good ideas and we'll revisit soon.
I think we'll be able to get that into v2021.3, thanks. Please also note that it's possible to use access controls to hide the entire Server category. This is something I've sometimes done without even setting an admin password just to reduce the risk of mistakes. Even if users could technically change the access settings back, in my experience they typically don't.
I think we're ok on this front because only the domain is used as the project name. So if your server URL is https://my.cool.server/secretsecret, you'll just see my.cool.server.