I have had also posted the issue in my own thread without any moderator response.
my report
I would like this to be picked up as a bug request for AppleTV App. No other playback method is experiencing this. I have a FireTV stick which is fine. Both AppleTVs stutter after a while. The more often you launch a new stream inside the app, the worse it gets.
You can easily circumvent the stuttering by “killing” the process on the AppleTV and have a fresh start of the App. This usually results in a shutterless playback. If you stop and launch a new stream it will start to occur after a while.