Re: The law of Unintended Consequences applies....
I think you are aiming a bit high in this instance. Let me use Youtube as my example here.
If you give Google/Youtube full responsibility for everything published on Youtube, it will be shut down before the ink would have time to dry as you correctly stated. This will also lead to an angry mob trying to lynch you probably including me as I no longer really watch ordinary television, but I do watch a large amount of Youtube videos as the channels I watch cater to my specific interests which no other media really do. While I would not cry for Google there are no alternatives as anything remotely like it would suffer the same fate. The exact same thing would happen to every social network (I use none so I don't care, but I think that horde would be even bigger and probably even have some torches and pitchforks in it!
The fact is that every single platform for non instant communication would be gone.
The reasonable thing to do is to require the platforms to make a resonable effort to police the content provided on it and force them to act on obviously illegal content brought to their attention. If they sling ads that would not be ok in another media then it should not be allowed for them.