YouTube runs fine for everyone else through Steam's Browser. Like six months ago, I was watching a livestream, and the guy was playing music by running YouTube in his Steam Browser. Try updating Flash and all that.
http://get.adobe.com/flashplayer/
>Click link>Your Google Chrome browser already includes Adobe® Flash® Player built-in. Google Chrome will automatically update when new versions of Flash Player are available.
Temporarily go there with Firefox or Safari.
>Already done that>Download it>Still doesn't work>:c
It always works for me. :o