Hello AG3NTEM,
Thank you for reaching out in Apple Support Communities. We recommend to ensure that the Apple TV is updated to the latest version of tvOS 16.3. This helps to provide software improvements that can help you to successfully sign into iCloud. If needed, here's how to update the software on an Apple TV:
"Update your Apple TV manually
- Go to Settings > System > Software Updates and select Update Software.
- If there's an update, select Download and Install.
- Wait for your Apple TV to download the update. Keep your Apple TV connected and plugged in to power until the update is complete.
After the update downloads, your Apple TV will restart, prepare the update, then install it. When the update is complete, your Apple TV will automatically restart again."
Following these steps can help and is outlined here: Update your Apple TV - Apple Support
If that doesn't help, we recommend to contact Apple Support to review this further. You can contact them by using this link here: Apple Support
Best regards.