This issue still exists in iOS9 beta.
I also have El Capitan installed and airplay works fairly well. It's only when I switch to my iPhone 6 plus or my 1st gen iPad mini (both iOS9 beta) that the struggle begins.
I can often get it working by turning airplay on first, sliding the switch to turn on mirroring, and then turning on app I want to airplay. Sometimes I have to play with a bit by turning mirroring on and off a few times.
The problem is I would prefer not to use mirroring, it changes how video is displayed on my TV. Without mirroring I use the entire screen on my TV, not so with it on. Now sometimes I can switch mirroring off and it will work, but it's touch and go, and often what happens is that it will start "stuttering" like it's constantly stopping and starting. Thats when I switch back to mirroring.
I mainly use it for my sling player, however it's even worse with apps like CBC TV, which has ads placed periodically in its video stream. Every time you hit a commercial, airplay gets screwed up.
Interestingly, when I use the Youtube app and "airplay" to the TV with the built-in wifi-direct it works great. How google can get their apps to "airplay" better than apple is beyond me.