Been lurking in this thread for a long time, as I was dealing with the same disconnection issues after upgrading to the latest 4K model a couple of months back. The remote would lose connection after anywhere between 2h to a day. Only fix was to reboot the Apple TV. I was getting similar issues with connecting Apple Watch for Fitness+ workouts, which kind of suggested a general Bluetooth issue (also fixed after a reboot).
Anyway, finally bit the bullet and called Apple on this. They eventually told me to “update and erase”, i.e. wipe the thing and start fresh. I rolled my eyes, because it’s such a pain to do, and clearly wouldn’t change anything. But I did it.
Guess what? After more than a week, the thing hasn’t disconnected once. So far, so good. It seems rock solid.
TLDR: if like me you’ve been driven crazy by this issue, try a full erase and update - it seems to have worked for me.