I thought it worked like chromecast and handed the URL off to the Apple TV which played directly, with the host keeping remote control for play/pause etc? The way you describe would be really inefficient. Mirroring your desktop would be different of course.
I believe that is how it works for iOS devices, but I am not so sure from a Mac. When I use the AppleTV as a 2nd screen I can drag the video from the laptop's screen to the AppleTV and it never skips a beat. If it was streaming from the AppleTV after dragging it to that screen I would think it would have a hiccup, at least, as it switched from streaming from my laptop to streaming from the AppleTV, no?
EDIT: OHHH! I just found you that you CAN AirPlay from a Mac with El Capitan, but you need to be using Safari or the Quicktime player. For example, when watching a YouTube video (in safari) it has an AirPlay icon at the bottom. I tried Twitch, but no luck, no AirPlay option. Link to Apple's Site