Sign in with Apple not working for new apps?

Since a few days, we are hitting AuthorizationError 1000 / 1001 whenever we try to use Sign in with Apple for a new app. We have added entitlements to the app in both release and debug, and setup the services id.

Many other devs are complaining about the same issue lately, and it's possible to reproduce on a fresh app id, see this thread:

https://www.reddit.com/r/flutterhelp/comments/1lf6kmq/sign_in_with_apple_signup_not_completed