It would be great for videos to default to a permissive license, but I don't see how that would change anything fundamentally with regard to copyright problems.
You'd still have to deal with people uploading whatever they want, and so you'd still have to deal with DMCA takedowns, and you'd still have to deal with the vast gray area of fair use (remixes, snippets of songs playing in the background, parody and comedy, etc).
The point is it is not ambiguous. I think the hardest part with video content is trying to figure out the boundary line. But like I said, whether or not you have copyright assignment rights is really straightforward.
It is ambiguous. Sure, if they upload a clip of a movie, that's not ambiguous. But fair use certainly exists, and you certainly have copyright assignment rights on works that appropriate other works as long as it fits the fair use criteria (or are sufficiently transformative).
And you'll still have hordes of unhappy users when you take down their Let's Play videos and videos of their kids when they had Prince playing on their computer in another room of the house.
You'd still have to deal with people uploading whatever they want, and so you'd still have to deal with DMCA takedowns, and you'd still have to deal with the vast gray area of fair use (remixes, snippets of songs playing in the background, parody and comedy, etc).