> When you start an Airplay stream on your phone directed at an Apple TV, the Apple TV doesn't send a switch input command.
Oh? My Apple TV 4K (2021) turns on the TV and receiver when I start AirPlay, and the receiver switches the input to the AppleTV if it wasn't already on it. This works whether the TV+receiver were on or off. My TV is a Sharp from ~2013, and the receiver is a Yamaha also from around the same time.