Currently you have many answers where people reply to original answers, ask clarifications. Then people reply to the replies. Because of the different sorts that are available, these become extremely confusing and hard to follow. I think that back-and-forth conversations of this sort can be very useful for discussing specific topics (example: users want to discuss one aspect of a suggested answer). With the current functionality, this just is very hard to do.
Suggested feature: one level of threading is available for answers. A "Reply" button will be available below every answer for this. If the answer is a reply, it will always appear threaded beneath the original answer. First level answers will appear based on whatever sort the user has chosen. 2nd level (reply) answers will appear indented beneath the original answer, always sorted chronologically (oldest to newest). I think that this will allow the current answer system to function to its best potential.
I recommend using chronological “oldest” sort — if there are that many conversations going on, the question is flawed. most of the answers should be about the question, not replies to each other.
newsvine is a good example, but I'd argue it's not quite the same thing we do -- it ends up being about the commenters and not the question. In other words, a news article isn't a question.
deferred for a few months
...from whence they were spawned.
Check out Newsvine (sample: http://tinyurl.com/576l8n) for an example of threaded commenting that will hopefully not make you hate yourself.
What things "should be" is an ideal that I don't think is echoed in the reality of the site. Due to the nature of the site, I think that replies to answers will be more of a normal occurrence than you may have preferred. Especially when you have answers that are something like "I agree with what @User said, with the following caveat - I think that these should be directly connected to the reply...