I tried to stream a football game yesterday via the NFL GamePass app from my iPhone 5s (running iOS 9.1) to my new ATV4 - which is connected via ethernet to my network. (I have the latest Airport Extreme router on my home network). It "paused" (black screen on tv with spinning wheel) at the 30 minute point, then about 20 minutes later, then again. Each time I would turn off the GamePass app, restart airplay a few times, etc.
For the second half of the game I gave up on Airplay from the iPhone and fired up Safari on my 2012 Mac Mini (running OS X 10.10.5). Went to the NFL GamePass site, logged in, started Airplay to the ATV, put the Safari video window full screen, and it was rock solid for the entire second half (approx 2 hours). So maybe your issues are with El Capitan and not the ATV. Not sure what my iPhone to ATV issue is though.