Ok. I run Apple TV for Zwift. In the past I always had to use companiom app and I had to both “pair with bluetooth” and I had to “pair with phone”. This was necessary bc I had to in order to get all my functions working. In other words, to get the Kikr Bike, cadence, and HR I had to pair with phone. Recently I was doing same protocol and kept losing connection. I tried everyrhing! Finally I realized now I no longer have to “pair through phone”. I only have to pair through Bluetooth and Im no longer losing connection. Maybe I never was supposed to pair through phone and it was just working anyway. But I sure thought it was required due to apple tv only allowing 3 connections with one being the remote. All I can say is now if I DONT pair through phone my Apple TV works perfect without connection failures. Or at least it did tonight. Maybe this will help someone. I wish I inderstood better to explain.
You only need to either “Pair With Bluetooth” (i.e. with the BT on your ATV) or “Pair With Phone” (i.e. with Companion on your phone). You never need to do both because one will override the other.
As you say, the ATV only supports three BT connections, including the Apple remote. Since I need to pair three BT devices (my trainer with power/cadence resistance, my HRM, and my Zwift Play), then I pair with phone (Companion).
If you only need one or two devices then you can pair directly to the ATV (what the screen calls Pair With Bluetooth).
Not sure if it will stick, but I went to my appletv settings and unpaired my phone from appletv. Seemed to fix it. My theory is that the appletv gets glitched out by bluetoothing a bunch of devices that my phone also Bluetooths (including each other)