There's an unsaid understanding that if you make a tweet, it goes to your followers. If someone RTs it, it goes to their followers. This understanding was broken by Liking becoming an action that could send tweets to followers' timelines. This made a lot of people upset, as tweets went to unintended audiences because people would like tweets and think that wouldn't do anything other than show appreciation. However, at the very least, there was some degree of interaction and you couldn't be some random person specifically looking for fights.
So how do random people find tweets that don't show up for them at all?
Present figure 1 below.
Figure 1 makes it so that anyone, anywhere, can find your tweet about [topic]. In theory, this is a great move because it ups impressions. In the real world, I'd guess that about half of the people using it are looking for people they disagree with so they can get in a fight with that person.
Figure 1 is also something that can be removed with little to no effect on people legitimately using the site.
You may think I'm joking, but I'm serious:
I have never seen a legitimate use for Twitter's search functionality.
Possibly the most legitimate use out of all of them, however, is Brands looking for people complaining about their service. However, even this gets abused -- brands will respond to tweets that were intended for only the audience they thought it would go to. So, instead, I suggest that people just tag the Brand if they want them to see it like any sane interaction would work.
Make it so that only hashtags can trend, and only hashtags can be searched. If people want to have a global dialogue on the tweet, they can tag it. If they just wanna talk about something to their audience, they can...simply not do that. It's that simple.
Just put a checkbox under Privacy settings that makes it so your tweets never appear in searches. You can announce it. You can tell one person. You can be completely silent on it. I guarantee, no matter what, the people who need it are gonna find it and you'll see a tweet with 24.1k+ RTs about how to opt-out of being searched.