I would ask this at the trainers club site, itself; but their “support” is nothing more than a FAQ, and it doesn’t address this question.
To start breaking in a new trainer I made a trainers club account and use that to sign into the game (since it wouldn’t let me create a different google-linked trainer). When I close the app for any reason and then start it again, the auto-sign-in waits seemingly forever, until I sign out and sign back in. This might be tolerable if it kept my settings between sign-ins; but I have to turn the sound back off and re-enable the battery saver every time.
When the net was bogged down, the sign in took a long time to complete using my google account; but it eventually completed if I waited a few minutes. I’ve waited an HOUR for this to work using the trainers club sign-in without having to sign out and sign back in, and it’s still waiting. If I sign out and then sign back in it will finish in 15 seconds.
If it’s going to require that, then it should at least have the close-app function sign the player out, itself, ask us if we want to sign out before closing, or preserve the settings. The way it is now is not user-friendly, it’s user-hostile.