Over here in germany we encountered a similar but different issue:
I'm an admin of quite a large school. We have around 1000 wireless clients (every student 7th grade and up and every teacher has an iPad plus several iPad class boxes with 25-30 devices each, smartphones and laptops, etc.) online at peak times and over 100 86" digital Boards in classrooms and several AppleTV's in office rooms. Teachers and staff use the Screen Snyc to connect to the boards and it all worked pretty well.
Suddenly, when Apple dropped their 18.7 and 26 iOS Updates on a wednesday Airplay went full bollocks. As soon as enough traffic hit the net, Airtime went up like crazy (over 75% on the network controller) and the boards weren't visible anymore in Screen Sync. Sporadically some of them were, but different ones each time you searched. Teachers couldn't connect to anything basically. In the breaks or later on a day all boards showed up again as if nothing happened.
There were no updates on network devices or the controller, there were no updates on the boards...just the iOS devices got them.
We tried every trick in our sleeve, we cut contact to Apple Update servers in our Firewall to reduce traffic, we tidied up our network, we tried several settings on our network controller for days, we even switched some network devices out because we thought maybe one was faulty/broken...until we found a solution:
Since mDNS and IGMP Snooping were active we added a "Querier Switch" and activated "Multicast and Boradcast Control" which prohibits devices from constantly casting their annoying and unwanted signals to everyone. Both settings weren't needed before. Then we put all important devices like the boards and AppleTV's in the exceptions. And boom, every board was happily visible and the connection speed even at peak traffic times is even faster than it ever was before.
So if you're in a bigger environment and you have issues with AirPlay, maybe this'll help you out. Our guess is that Apple changed something in the AirPlay protocol that screwed everything up a bit.