YouTube doesn't work in Safari on mid-2013 MacBook Air 13".
I've tried enabling HTML5 video at http://www.youtube.com/html5, and some videos did start playing after that, but most videos still don't work.
What I've noticed is that if you're viewing a video that does in fact work, and go into the full-screen mode, then all subsequent videos (which are suggested after the original video ends) do work, but only whilst you're still within the full-screen mode — as soon as you exit the full-screen mode, depending on the video, Adobe Flash might again be required right away, without any option of going back fullscreen.
How do I make YouTube work without Adobe Flash? Clearly YouTube videos should all work without any exceptions, since they all are already encoded in H.264 just fine.
Best Answer
So this isn't a total solution but it should help.
You reported that even though you joined the html5 program the following video did not play in html5:
http://www.youtube.com/watch?v=5NV6Rdv1a3I
I was able to reproduce that result in Chrome.
Here's how I was able to get the video to play in html5:
In Chrome:
View > Developer > Developer Tools > Click on the little gear icon
In the resulting preference window:
Settings > Overrides > User Agent [X] > iPad -- iOS5
Then reload the video and it will be in HTML5.
Similar solution exists in Safari, but I have to walk the dogs. Back soon.