What could be changed about twitter to make this better? The entire design feeds right into this. I've heard it said that it was intended to be a place for good discussion, but I really don't believe that. Good discussion isn't had 140 characters at a time. The whole point of the site is to give everyone a megaphone.
Use https://tweetdeck.twitter.com/ - normal chronological feed and no trending anything. Then prune your following. I don't see any witch hunts; my feed is generally very positive.
I have tweets come up in chronological rather than selected order, and it's still crap on a regular bases because people are crap. There's not much you can do besides unfollowing people who are consistently annoying or vapid.
I think retweets/quote tweets are a bigger problem. I can understand why a large number of people I follow might tweet about the same thing, but I don't need to see that thing reproduced 10 times.
> I've heard it said that it was intended to be a place for good discussion
What? No. Where had you heard that? Twitter was meant to be SMS in the Internet - for quick comments, quick news, quick updates, short notices, short etc.