Unless you're going to throw out local storage and custom request headers, getting rid of cookies isn't really going to do anything except make the same thing less secure (since you won't be able to benefit from the HttpOnly flag).
Typically the user agent will redirect to a Google/etc login page where it'll have access to first-party cookies. Then will redirect back to the site which requested authentication, passing state in the query params. It's only when you get into using stuff like Okta as a delegated authentication service do you run into trouble with 3rd party cookies.
Edit: As an example, I just logged into Stackoverflow using Google as the authenticator, with umatrix blocking 3rd-party cookies. Worked without a hitch.
The user clicks 'log in with google', their browser gets forwarded to whatever.google.com, the (now first party) cookie gets checked, then the user gets forwarded back to your site with the access token as a parameter in the GET request.
No third party cookies needed.
I block third party cookies indiscriminately, and I never had issues logging into Facebook, Google, Twitter, or any Microsoft service except for Microsoft Teams. This includes logging into third-party services using the Google login.