With iOS 17 and tvOS 17, I am seeing the Apple TV status in Home update itself within a few seconds of taking an action.
"Playing" when something is playing on the Apple TV
"Not Playing" when the Apple TV is turned "off" (I actually had to also force quit the app I was using) EDIT: I just notice a "Stopped" status here too. No idea what the difference is between the two.
"Paused" when I was playing something and hit 'pause' button. (seems to stick on this even after turning off Apple TV unless you force quit the apps)
Not perfect, but it's better than before.
IMO, much of the confusion revolves around this concept of the iPhone and Apple TV not really having a "Stop" button. When you play something, it gets "loaded" into the "deck" (my terminology) and you can only "pause" it. Sometimes it remains there for days, or until you "replace" it with something else. It's "booger-ware" and I think it's a pain in the ass... it's what causes a song from two days ago to start playing in my car when I have not hit play. There needs to be a hard "STOP" function that CLEARS out the deck. Force-quitting the Music app has no effect on the contents of the deck either... booger-ware.
EDIT: I now just noticed a "Stopped" status that I cannot explain. When OFF, it shows either "Stopped" or "Not Playing". What is the difference? Does Apple document how this is supposed to work?