This is one of the weirder bugs. I have been heavily using Music on my 4K ATV for many years. On most software releases this bug (not activating the sreensaver even if correctly configured) has been there, going back years. I had a short period in the early tvOS 16.X versions where the screensaver worked as it is meant to, now the bug is back on my current 16.6 version. It somewhat beggars belief that this is so difficult to get under control, given it gets reported pretty much monthly by somebody.
I now have the cover box, and more strongly the place where the song name and artist is displayed in white, burned into my OLED screen.
As an aside, I have had a similar experience with the ‘volume control via another device’ feature. It used to broken for a long time, then worked for a while, and it is now broken again.