I don't think it's fair to lump Twitter in with Facebook. Sure, if your feed isn't curated or you're looking at the replies to a Trump tweet, it's going to be a dumpster fire. But otherwise it can be a tool for keeping people informed without having to rely on the news.
It does require a critical eye, which the majority of people don't have, but I'm not sure what the best way to solve that problem is. Moderation at that scale doesn't seem feasible, and a downvote system to silence people, as you suggested, would be easily abused.