You make excellent points. Twitter IS for broadcasting. It is the counterpart to community interaction. Say, you want to reach those who aren't in your community. That would be difficult on Facebook. In fact, it's a "uniquely defining feature" of Facebook not to facilitate that sort of usage. They both have a purpose. Say, a startup needs to communicate app status to its users so people aren't left hanging in mystery during a downtime. Twitter would help with that well.
Maybe Twitter IS inherently less warm and more chaotic. I get that feeling too. (I don't even log-in often.) But I sort of use it to create a pseudo-community.
Here me out.
140 characters are limiting, right? That means that the author of a post has two options: risk losing followers by posting chatter, OR make sure each word matters so that it's a saturated snap-shot of their current thought.
Now, imagine that you follow a curated list of influencers (i.e., people that are involved-in and doing things you also are passionate about). THEY may not all know about each other, but YOU know about all them. (Think: one-to-many instead of many-to-many.) As a consequence of this, you're continually getting an influx of musings by people you respect and value the opinions of.