*I do have a once-per-run Moonlight glitch that may even be because I'm on an older version. The first connection just doesn't pick up on the HDR immediately. I just have to reconnect once and it's good after that.
[1]: https://github.com/ClassicOldSong/Apollo ("Built-in Virtual Display with HDR support that matches the resolution/framerate config of your client automatically")
I thought I surely was imagining it, so I revisited it 6 months or a year later, and, yes, the input latency from the controller is substantially worse on the Apple TV.
I had unacceptable input lag when streaming to my Apple TV 4k, but switching the TV it into "game" mode fixes it entirely. I just need to swap it over to game mode when I hop into Steam Link or Moonlight.