The whole reason Stack Exchange exists is to let people post questions and answers to those questions. Since the very first day of Stack Overflow's existence, Qs and As have been considered first-class content.
Comments, on the other hand, are widely considered second-class content, and have been that way since shortly after they were introduced, early on in SO history. Terms commonly used to describe comments are "temporary", "ephemeral", "throwaway" and the like.
Why, then, can regular users delete questions and answers but not comments*?
Currently, when it comes to deleting comments, the entire burden falls on moderators. No amount of rep allows non-mods to do anything more than flag for mod attention. One of SE's core principles is giving the community authority and ability over as much moderation as possible. Adding the power to delete comments would represent a good step in that direction, not to mention take some pressure off of our hard-working volunteer mods.**
At this point, some of you may be thinking this is unsafe, because regular users can't view deleted comment history, so there's less oversight than for deleted posts, which trusted users*** can see. However, as already noted, comments are inherently unlike posts. It is by design that they are second-class; stuff that's too important to lose shouldn't be put there in the first place. So, oversight has reduced importanance.
Another possible argument I can think of against this would be that comments don't matter enough to merit a rep-based deletion privilege. In other words, giving people this privilege would take up more of their time and energy than the comment cleanup would justify. Obviously, I don't agree with that argument, but I can see how someone else's cost-benefit analysis might differ.
(There's also "insufficient dev resources", I suppose, but that's a practical limitation, not an argument against the idea.)
Implementation of comment cleanup would improve the all-important signal-to-noise ratio on the sites, and decentralizing this ability would let problem situations get quicker responses—"deleting rude comments helps stop fights before they begin" -Brad Larson—by increasing the number of (empowered) eyeballs around.
To make this a concrete proposal: can we give trusted users*** some ability to delete comments?
*: granted, non-mod post deletion requires multiple people who are each above a high-ish rep threshold and applies only to posts meeting some low quality prerequisites, and non-mod comment deletion is available in a few special cases like offensive content or thank-yous
**: I think I remember reading that comment flags make up a large portion of overall flag volume, but not where. If someone can find a source either way, please leave a, um, comment.
***: AKA 20k users, AKA users with delete powers