Oddly, their live streaming does not work without Adobe Flash. It absolutely refuses to load anything in Chromium, even though it supports every single HTML5 format that YouTube offers. I would have expected better from Google in 2014.
That's a very bad way of stating it. It's perfectly possible, and "supported" even though there is no "use this thing" solution for it. The problem is that you can't rely on a full solution being usable by all browsers you want to target.
Why hasn't HLS caught on more broadly? From what I know it's not licensed / patented by apple. Many Flash players implement it, and overall its a great way of delivering video over HTTP.