There's a lot of variety in human brains. What will traumatize one person is comedy for another. There are (or at least used to be) websites you can seek out to look at pictures of grizzly death. It's simply not the case that everyone is damaged by looking at terrible things, and - to cut off that line of thought - the people who aren't are not monsters or sociopaths. They're just made of different stuff. I can't comment on whether Facebook et al are careful enough in their hiring practices, but I'm certain that there are plenty of people in the world that can do this job.
Email lists were popular before Facebook took over that niche. Nobody more than I would rather see the comeback of mailing lists, but that doesn't solve the problem. You can assume that if mailing lists got the kind of uptake that Facebook has, gmail/hotmail/etc would have the same content curation issues.
I'm still not sure exactly what you are proposing. Facebook gives me a way to communicate with my friends. It sounds like you want to take that away.