You're just flat out wrong. That's why there's a tab dedicated to replies. Twitter conversations can be incredibly interesting and benefit all participants by being public. Twitter's setup hugely encourages @ing people in your tweets.
Except conversations are badly broken. So many people don't hit the actual reply, so conversations get cut off, often restarted, and thus become fragmented, which makes them hard to follow and participate in. Sometimes it works. Often it doesn't. I think this is what the OP was talking about.
But this is not a problem of the platform, but of how users are using the platform. I rarely have conversations on twitter (i.e. most are just a couple of replies long at most) but I occasionally have a 20+ post. If you hit proper reply, it works. If you don't, it doesn't (or does really badly), but this is a user fault, not a platform fault.
If users have difficulty properly using the platform, the platform is at fault. An arbitrarily naive user should never have to fight the platform to do what he wants.
Also consider when there's say four people tagged in a conversation, and then someone needs the extra characters to make their point and they remove one of the names, then that person loses the context too.
Also, conversations look different when you're looking at them from one of your lists, or if the people you are following are different than the people I follow.
So many clunky issues that you need to be on top of for it to make sense, and this is what's lost on the general person.
This is one of the many examples of Twitter not knowing what Twitter is. Twitter is a product that was a success in spite of the efforts of its founders.
