I have the same issue, and did uninstall the Apple TV app, and did a full reboot/restart of the TV after the uninstall and then re-installed the application.
I continue to experience the same issue.
The Model is compatible, (7137x), it is not using an SD card, it pulls in content so it’s connected, the time is set on the Roku TV and the AppleTV 4 standalone currently connected to the same network and same TV is fine.
Attempting to use “Sign in on mobile device” generates a nearly immediate “Verification Failed”.
“Sign In On This TV” after putting in Apple ID and password results in an endless “Signing In”.
The Apple ID and password are correct, every other app on the TV has connectivity.
Seems like the auth service can not connect as configured.
I have uninstalled / restarted / re-installed several times with the same result.
Can also not change “Location” to “On” under privacy settings but easily able to change Show Sports Scores to On or Off.