Try restarting the router, optional other network gear, and the TV. Not sleep/wake, but really off power.
Use regular ISP-provided DNS servers on your router, and have your devices use the DNS as indicated by the router (“Automatic”).
On the TV: Check for updates for the Apple TV app. Check for updates for the smart TV OS and/or TV firmware.
Verify your time and date and time zone on the TV. Software relies on reasonably accurate time for secure connections (TLS and the like). Getting your library will require a secure connection.
Contact Apple Support if you need additional help with accessing your library.