Oauth logins without email as an option

This seems like a really bad idea to me: offering only (presumably) Oauth for registering for a new service, with no email option. Does either the website, or the user / customer, really want to be dependant on a third party company for authentication?

If Oauth was exclusively used, and the website itself was an Oauth provider, it wouldn’t be so bad (at least the website has control over its own authentication). But relying on Google, Facebook, Twitter, Apple? Google services are probably “illegal” in the EU right now. Not a good start!

Shame because I love Power Thesaurus and use it often.