Register extra router from the authmanager into the FastAPI application

[pierrejeambrun]

Body

Depends on #44847

The extra router exposed by the auth manager should be registered into the core FastAPI application.

Committer

• I acknowledge that I am a maintainer/committer of the Apache Airflow project.

[pierrejeambrun]

cc: @vincbeck

[jason810496]

Hi @pierrejeambrun, I can work on this issue, could you assign to me? Thanks !

[pierrejeambrun]

Hello @jason810496, of course, thanks for your involvement, assigned.