Google has just launched a fix for this - YouTube EDU. The educational version of YouTube has the comments and related videos removed, keeping the focus on the video that the teacher wants the students to see.
It does take away the exploration of new videos through the "related" videos part, but I think teachers will be ok with that. As for no comments, teachers can set up a blog or other site for students to comment on videos.
There is content from over 600 educational groups, including TED and the Smithsonian. They have even created playlists for the teachers.
School administrators can enable access only to YouTube EDU so that teachers can access the educational materials on YouTube.
This is a great resource for educators.