On https://github.com/flutter/packages/tree/main/packages/google_sign_in/google_sign_in_web under the "Usage" section, the example link refers to the old google_sign_in package.

This documentation page starts with "Migrating to v0.11 (Google Identity Services)" and this package (google_sign_in_web) was created (or updated?) to implement sign in using the new GIS service.

At the time of writing this issue, I could not find any example (tests do not show the intended flow) that demonstrates how to use this package in a simple SignInDemo (similar to the old example).

Maybe it is already work-in-progress looking at this scaffold https://github.com/flutter/packages/blob/main/packages/google_sign_in/google_sign_in_web/example/lib/main.dart