Q: Web Video is streaming correctly, but doesn't play?!

Hi. I'm not sure if what you quoted is working for you. The correct solution is to AirPlay the video as if you expect it to work. Then, after it starts downloading to ATV, open Control Center and briefly turn AirPlay off and back on. This seems to trick whatever logic is failing. You should let Apple know here https://www.apple.com/feedback/appletv.html.

Yes, I know the issue is present for you when on websites. I was asking if it's happening when you airplay using an app. Does it also occur when tested on another network?

Just for reference, this started with tvOS 9.2. AirPlay was working fine until then. The first discussion, including some false starts is Re: AirPlay Won't Play Videos Well.

Take it to Apple

Yes. The trouble is that it's intermittent, and doesn't affect all videos. Given the nature of the fix, it seems like a timing problem between different software components. And, as you've observed, some sites never fail. You should report it here https://www.apple.com/feedback/appletv.html.