
Here’s the official support article for the procedure:
If your TV shows an iPhone next to an Apple TV - Apple Support
It will try to “recover” (reinstall parts of?) the system software through your iPhone’s connection. Apparently that is needed to continue. This may take more than a few minutes. It is a great method if it works (as you can do it yourself) — if it works.
In your video, it also seems like the bluetooth connection and/or Wi-Fi gets lost for a moment.
Verify that there is no interference from other radio sources in the bluetooth radio band (~2.4 GHz), nor Wi-Fi radio bands (~2.4 GHz, ~5 GHz), nor shielding (often metal) in between.
Resolve Wi-Fi and Bluetooth issues caused by wireless interference - Apple Support
If the issue persists while optimizing the environment, then please contact Apple about this:
Restore your Apple TV - Apple Support
“… your Apple TV 4K might need service. ➔ Get service.” (ask for service, not for a repair!)