OpenID still required sites to implement that standard as a relying party so it couldn't be used arbitrarily. It gave you the ability to choose a specific OpenID identity provider to store your account info (including your own), but it didnt get adoption so now it's obsolete.

OpenID Connect is the latest version of OpenID, and it uses OAuth to federate access to other identity providers but sites now have to implement a separate registration for each provider (google, facebook, etc) so what they choose to implement is what you get, and there's no realistic option to use your own.