Good questions. I've come to believe that Airplay isn't as straightforward as it seems it should be. You'd think it's always "just" sending the video and audio data stream from your phone (for example) over your local network to the Apple TV. However, from what I've read, in certain situations when Airplaying a display from an iPhone app it acutally sends a URL to the Apple TV, which downloads the video itself (i.e., the phone is not involved after the handoff of the URL). In this latter case it could result in better-quality video than if the phone were rendering the video and sending screen data to the ATV, and could be less-demanding of your WiFi network. Could this be happening? From what I remember this only happens when Airplaying from an iOS app that has the Airplay function embedded in the app... but I'm not sure.
What's the source device? Is it on WiFi or wired ethernet? How about the ATV -- is it WiFi connected? How are you testing display sharing -- from within an app, or from control center in an iOS device, or a Mac? (Keeping in mind the claims I make above, maybe try display sharing in various ways to see if it still works fine in every mode.)
It could be that the WiFi signal isn't "good enough." (I've had plenty of instances of dropouts while doing audio Airplay over WiFi.) Any chance that you can connect the ATV to an Ethernet cable, even temporarily? (I know you said display sharing works fine, but if an Ethernet cable fixes the audio dropouts, well, that probably tells you what you need to know.)
It's great when this stuff works, but sure is frustrating when it doesn't!
