If I'm authenticating with server A. I shouldn't have to carry ephemera from server B. A can interact with B on its own if necessary.
Bubbling up these architectural details to the front end is a symptom of the webdev cargo cult coming up with broken ideas that get fossilized as the status quo.
With OIDC, both occur: the client is redirected to the authentication server where they directly authenticate, then carries a token cross-domain back to the service. Finally, the service validates the token against the auth server.
The alternative would be something where I enter my Google username/password on random websites, and trust that they will forward it to Google and not do anything nefarious. This is less secure and less private.
Bubbling up these architectural details to the front end is a symptom of the webdev cargo cult coming up with broken ideas that get fossilized as the status quo.