I would say that the answer to this is dependent on the situation.
At least judging from my (somewhat long) experience with this stack, it allows for somewhat more humor/wit than most, even when it's not directly related to clarifying or suggesting an improvement to a post. Personally, I like that. I don't personally feel that a witty comment or two on a question or answer hurts anything and I don't think deleting them helps anything in most cases. On the other hand, if it gets to be more than a few on a given post, at least some of them should probably be deleted. This is, of course, assuming that the humorous comment in question isn't offensive. Comments that violate the Be Nice policy should be deleted immediately regardless of humor.
Regarding the witty/humorous comments, at times, I think it can be a subjective judgment call on which ones get deleted. If there's a lot of noise in a given comment section, removing all of the humorous comments may be useful. On the other hand, if there were one or two that I found particularly witty and/or which attracted a lot of upvotes, I'd personally prefer for those to remain. I don't think having a sense of humor on the site is a bad thing.
Relatively Brief Discussions of Relevant Information
There have been lots of times on this site where a comment posted on an answer has resulted in a relatively brief and usually informative discussion of that topic, usually either to clear up a misconception of the comment's author or of the answer's author. Again, if these are kept relatively brief (say, no more than 4-6 comments) and they're informative, I think it's fine to leave them unless one or both parties desires for them to be deleted or the comments on that answer are getting too long.
In some cases, the answer's author can edit the answer to address the issue mentioned in the comments and, in those situations, it's preferable for the answer to be edited and then the comments deleted. However, in other situations, the side issue may not be directly relevant to answering the question and, thus, might not belong in the answer. In the latter set of cases, I think it's fine for these to remain in the comments.
One problem that can arise when these comments are moved to chat is the one that Koyovis mentioned in a comment on another answer here: the chat room can end up being deleted. As Koyovis pointed out, this answer is a good example of that. As Federico pointed out in the comments, these rooms get automatically deleted if they do not contain more than 15 comments by at least 2 users after a period of 7 days of inactivity. A user who was very knowledgeable in the subject had posted comments explaining why a given answer was wrong. The comments were moved to chat, but the chat link is now dead, since the thread contained less than 15 comments. In this case, it was a user who flew the F/A-18 discussing the launch of missiles from a fighter, but now his comments are gone (and he is unfortunately no longer an active user.) Unfortunately, moving the comments to chat was not useful in this case.
When the discussions start to get truly 'extended' (say, greater than 4-6 comments in a discussion,) then I think there's a better argument for moving them to chat. At least on other SE's, though, I've heard that comments on a given post can only be moved to chat by a mod once. (Someone feel free to correct me if that's not accurate, but I seem to recall reading it from the mods of other stacks.) Given that, I'd recommend that the mods be careful about not moving the comments so quickly that a bunch more end up being posted with the mods then having no choice but to leave them or delete them.
When an extended discussion does need to be moved to chat, I would also recommend that the mods be a bit judicious about which comments are moved. For example, if there's a comment with a given valid criticism of an answer which then sparks a lot of discussion, it may be better to leave the original comment on the answer even after everything else is moved to chat. This way, others who don't bother following the link in the chat will be aware of the potential problem(s) with the answer. Of course, if the answer is actually edited to address the issue(s), then there's no reason for the comments to remain.
The Nuclear Option
(OK, I just liked that title.)
Regarding the nuclear question in particular, the comments on the question seem to have a low signal-to-noise ratio. A few contain interesting information, but, overall, it's not very focused and, at least in my subjective opinion, none of them really stand out as a comment that should stay. I'd not personally miss them if a mod nuked them all.
People Flagging Comments Just to Flag Comments
In a comment it was mentioned that
there are people (mostly coming through the HNQ list) that actively go around to hunt for comments to flag for removal, and those don't even get moved to chat.
If the comments truly are useless, I can understand this. If the comments are not useless, then this behavior doesn't seem very useful and, personally, I'd recommend rejecting the flags. Dogmatically enforcing rules that were created primarily for SO just for the sake of enforcing rules is not useful. And it's even less useful when it's coming from people who rarely even visit Aviation anyway and are only here because of an HNQ question. If someone is going out of their way just to find comments to flag, I wouldn't feel too bad about rejecting their flags in hopes that they'll find something more constructive to do instead in the future.