The one-play-per-launch pattern is the clue: the first playback session starts fine, so this is not the files or your account outright; the playback/authorization state is failing to reset for the second item. The fixes in the order I would try them:
1. Refresh the media sign-in. In the TV app, use the Account menu to sign out of your media account, quit the app, reopen it and sign back in. Protected playback keeps a session token that can go stale exactly like this.
2. Re-authorize the computer. Account > Authorizations > Deauthorize This Computer, then authorize it again from the same menu. Second-play failures on purchased or rented content are a classic symptom of a damaged authorization, and this rebuilds it without touching your library.
3. When the gray box appears, a workaround that avoids restarting the whole app: open Activity Monitor, search for VTDecoderXPCService, quit it (it relaunches on its own), then try playing again. That is the shared video decode process, and a hung one gives you a player window with nothing in it.
4. To find out which side of the fence it lives on, try the same two plays in a fresh user account on the Mac. If the second play works there, the fault is in your account's TV app data; if it fails there too, it is system level, and a Safe Mode start is the next test, because third party display and capture drivers (DisplayLink docks especially) are known to interfere with protected video after the first stream.