Datapoint: I have a NEO2T and use Zwift on AppleTV most of the time myself. Until last week, that was a 2017 ATV4k. This weekend, I got the new ATV4k 128GB 2022 version. It’s running on both.
Uninstall the Zwift app on AppleTV. Also check the ATV for OS updates. Reinstall Zwift app. There was at least one recent report of an ATV Zwift issue solved doing the above.
If that doesn’t work, refer to this thread:
The dummy was me. Don’t be like me.
EDIT - the nuclear option is a complete reset of the ATV. That’s fixed one obscure issue BLE issues for me in past so I’d do that before doing much else.