There's a fundamental difference between curating content like a newspaper does vs what Youtube and similar sites do. It's a whitelist vs blacklist approach: a newspaper approves things one by one and only has so much content, while Youtube allows anything by default and then removes thing in certain categories.

And even newspapers use the latter approach for some things. I doubt the New York Times wants to be held responsible if someone posts a link to pirated content in an article comment, for instance.