I'm sympathetic, but you're going to have to define some thresholds.

If I set up Paul's Video Hosting Service next week, and someone wants to post videos of kittens being tortured, I'm not hosting their videos. I'm not even hosting videos of a presidential candidate making certain pronouncements. No US court would agree that I should be forced to do so.

To whatever extent it is reasonable for YT to be forced to host any legal content, where's the boundary between the putative pvhs.com and youtube.com?