It's a trust problem. It's not necessarily that they don't exist, it's that they don't trust you and they certainly won't trust the spoofed ID your own alternative login server is giving them. They aren't public, they're effectively private. That's exactly the point of having any auth at all.
The point of things like authlib injector is not for servers to blindly accept any arbitrary "alternative" auth that can be injected. That would be functionally no different from not having authentication at all, and then you get all the same problems that allowing anonymous, unauthenticated access allows: griefing, ban evasion, cheating, trolling. An arbitrary alternative auth server IS essentially anonymous... because one griefer could easily run a hundred different self-hosted ones that cater exclusively to them and their griefing friends and when you ban them and their auth server they just start a new one. The barrier to entry (and re-entry) is far too low to be practical. Nobody is going to do that, at least not for very long unless it's part of some intentionally chaotic experiment.
So, just like with non-authenticated offline mode servers, the only viable solution is whitelisting and strict moderation. Ely.by seems to be the only public auth server that I can find, and it is very difficult to confirm any public servers that explicitly support it. That doesn't mean they don't exist, but I suspect they are mostly limited to communities where even if the community itself might be reasonably open to public, you still need to be part of it before you get the details and instructions you need to access to their server.
authlib-injector exists so you, and your friends, and your community, whatever it may be, can have their own minecraft server with your own auth which you control, and that auth will not be optional or you wouldn't bother having it in the first place. It specifically won't be public, that's precisely why you're asking people to authenticate using your auth service.