I believe this can quite easily be done. The only issue I can see is adoption. Websites are not gonna spend time and money on adding authentication by an other

I believe this can quite easily be done. The only issue I can see is adoption. Websites are not gonna spend time and money on adding authentication by an other third party if this authentication system/service is not accessible by a huge amount of users. Like you said, most of the time what you can see is auth with Facebook and/or Google. That is because a huge amount of internet users have an account on one or both of those platforms and that makes it worth the money to add them to some website's auth system. On the user's side, some users will agree to pay for a privacy focused auth system but that will still be very few users. This can be an attractive solution for a subset of the internet users who are more privacy aware. But, some users will want services to use the auth solution before they create an account. It's a bit like an egg and chicken dilemma.