TL;DR Please, allow very-high rep users (50,000 rep on "full" rep levels, 10,000 rep on "beta" rep levels, and 5,000 rep on "private beta" rep levels) to help review No Longer Needed comment flags.
I'd imagine this could work by adding it as another review queue, where the comments are sent to after they are flagged. It would require 2 net votes to delete from high-rep users in order to actually be deleted (and leaving the queue at -2 net votes to delete). In order to prevent problems, users could not review comments if they flagged the comment, commented on the post, or made the post. Perhaps, in order to help get rid of other useless comments, it could show all the comments under that post.
No Longer Needed comment flags are used to flag comments that are, as the name would suggest, not useful. These types of comments flags tend to be pretty easy and obvious (not just my opinion here, this seems to also be the general moderator consensus), and yet only moderators can deal with them (unless it happens to be nuked by enough regular flags, which only happens 0.1% of the time). On Stack Overflow alone, in 2022, 243,198 comment flags were handled by moderators. That's nearly as much as how many flags on actual posts were handled my moderators in the same time. In 2021, that number was 403,458, significantly more than post flags. Based on this data, about 71% of those were No Longer Needed (NLN) flags. So, the Stack Overflow mods alone handled an estimated 172,671 NLN flags in 2022 and 286,455 in 2021, which could have easily been handled by the community, comments which I would imagine very rarely require anything other than a quick deletion.
As of right now, I'm not suggesting the handling of Unfriendly/Unkind, Harassment, Bigotry, or Abuse, or Something Else comment flags as those are much more likely to more difficult to handle and/or require moderator intervention. The numbers 50k/10k/5k and the 2/-2 are just ideas of number that seem roughly correct, I'm not particularly attached to the numbers here. The 2 votes to delete comes from the fact that it currently take 3 flags to nuke comments, and 2 more user (plus the original flagger) reaches that 3.
Yes, on certain sites, this review queue might get backlogged. But let's say that only 10% of NLN flags are handled by high-rep users because there are very few queue reviewers, and most still have to be handled by mods. That's still tens of thousands of flags on SO alone per year that mods now didn't have to handle. That's still a major improvement.
Yes, I'm aware of these other proposals for comment moderation, but I think that mine is significantly different and address some of the concerns brought up there. Yes, I'm aware that some of the smaller sites will not have 2 eligible non-mod users to do this (however those also tend to be sites with much less comments to moderate). I think that the queue should probably be turned of though for sites only 1 eligible non-mod user, as they would just be wasting their time. I think that we should probably make this an opt-in feature to also avoid the sites where there isn't really an existing comment moderation problem and/or there are very few users who could even review in it. I also think this should not be add on MSE and per-site metas, per the answer below. Yes, I'm aware that sometimes high-rep users make mistakes. However, we let users with much lower rep than this delete with only 3 votes actual posts, which are much more important than comments, and that mostly works out well. The chances of the original flagger, and 2 very high-rep users all wrongly thinking a comment should be deleted are rather low. Something similar to this already exists for chat moderation, and that also generally works out fine.
Sorry this question is so long, but there's a lot to say.
So, please, let the community handle these, and take a burden off of our moderators, and let them focus their time on the things that truly require a moderator.