I don't understand, say, sites like Facebook or Twitter are private companies, who let users store their messages/photos/videos for free (they earn the money in other ways) on their premises.

Good, but let's remember that it's still their property, they just let users use it, according to their own rules. The protection from lawsuits gives the tech giants more power with less responsibility.

I believe the world would be a better place if web properties took more responsibility of what happens on their premises. The internet would skew back to smaller web properties, and a centralized media giant would not be able to close anyone down because they don't like them.

Tech savvy people would make their own web properties, others would use it. It may or may not be connected by a federated communication protocols, but the important piece is that they own the property and be responsible of what happens there. E.g. if you invited your grandma to your forum -- be responsible of what she writes there.

PS: Of course, illegal content can still be closed down (e.g. through hosting providers and domain registrars), but not by a whim of a private corporation.