TL;DR: The reason for this policy about comments is that the StackExchange network was designed to be a Wikipedia-style, easily referenced Q&A database. On the other hand, this is nothing like what mathSE has actually become.
StackExchange is intended as a Q&A network. Among other things, this means that the most important questions are those that more people will have later, all information on a page should be kept as relevant as possible, and the most relevant information should be highlighted and accessible (accepted answers, upvotes). Irrelevant details should be removed not just from comments but also from questions and answers, and Wikipedia-style editing allows this to happen efficiently. Questions that are too localized should be closed under a strict adherence to this philosophy, because they are unlikely to help future visitors to the site or to show up on a google search result.
It follows directly from this philosophy that back-and-forth commenting should be discouraged. Instead, important information should be edited into the post itself, so that it is more accessible. This keeps the site from becoming like a forum, where you have to dig through pages of cumbersome details to find the information you need.
See, providing extended help in the comments is never appropriate.
And extended discussions are not what this network was created for. This is a Q&A Site.
What mathSE actually is: a forum. Sure, it poses as a Q&A database, and because of the site design a lot of the features are optimized for such a database. But at some point, the entire philosophy behind StackExchange seems to have gone out the window.
It is a forum because you ask any question you want, and can hope to get an answer as long as you provide context (notice the focus on context rather than on value or usefulness of the question itself). "Too localized" as a close reason is now effectively gone, with no real incentive for anyone to enforce it anymore and no clear indication that it should be enforced at all. Interesting, obscure questions get lost in the swamp of conceptual duplicates which have been answered hundreds of times before. Finding duplicates, editing, and closing are not rewarded by reputation so they are not particularly widespread behavior, and certainly they are not widespread enough to put a reasonable dent in the never-ending barrage of questions.
Worse, the Hot Network Questions list (coupled with some related issues) has made it so upvotes no longer indicate either the quality of a post or its widespread applicability. Instead, a large number of upvotes probably indicate that your post is a fun distraction or worthy of attention on Reddit.
Does this comments policy make sense for the site's actual (intended or unintended) purpose?
Many respectable users have upheld mathSE for what it is, and I do not fault them for it; they enjoy helping people, and they are not apparently too fazed by the unending stream of questions or the incredibly low bar we set for content. They may say that it is annoying or pointless to move comment threads to chat, and that it makes it harder to help people. Of the two positions here, theirs is certainly the more consistent with this site's actual usage.
On the other hand, the software wasn't designed for localized homework help. It was designed for excellent questions that help future users. And, it has served its purpose here to a large degree as well. I am often googling math questions and finding incredibly helpful answers from mathSE, formatted perfectly and free from irrelevant information. So it makes sense to at least try to uphold what the site was actually built for, and what in my opinion is really where most of its potential lies.
But those who are insisting that we continue to moderate comments in the way the software wants us to are probably fighting a losing battle, and the votes on this meta thread are not particularly encouraging. Perhaps we should cut the act and embrace mathSE for what it actually is.