I'm also having the an issue on latest AppleTV 4K and Xcode 13. Xcode's Devices and Simulators sees the device under the "discovered" list, and there is a "Pair with Living Room" button. (Living Room is the name of the AppleTV device).
When clicked, the pairing code does appear on the TV. I enter that pairing code in the Xcode Devices & Simulators pairing window. A Spinner shows for a bit, and eventually resets to show the "Pair with Living Room" button again. The device remains in the "Discovered" list, never reaching "Connected".
From the Apple TV side, the computer name does appear under the list of connected remotes and devices.
But since Xcode never completes the process, I am unable to develop or build to the device from Xcode.
I've tried removing the paired computer from the Apple TV's list of remotes and devices, re-pairing from Xcode, rebooted both machines... nothing is working. No matter what, Xcode just seems stuck in a loop where the pair never completes.
How can I fix this?