Just a quick note – I was struggling with Airplay “suddenly” not working from my desktop iMac 5k to my 4th generation Apple TV. When I would try to extend my desktop to the Apple TV, the TV would blank out and sound would start coming from the TV. No video would show at all – and I couldn’t move the mouse to what was supposed to be the extended display. I confirmed using a Macbook that the problem was with the iMac, not the Apple TV.
I searched quite a bit for a solution, and eventually found a mention that having other screen sharing tools installed might cause the problem. I had recently installed Duet and tested it (it worked!), so I figured that might be the culprit. After an uninstall and reboot, Airplay the Apple TV now works again. Hooray!
Duet looks to be pretty neat, and it does work (although the value of it on something smaller than a 12.9″ iPad Pro is questionable), but sending stuff to the big screen is far more important to me.