Thank you for replying Asaf. This is my response to your post. There's plenty of stuff here that I agree with to some extent. I do have some concerns though.
Firstly, I would have liked it if the community was privy to the discussion between the moderators about when to remove questions from the HNQ. This could easily have been a meta post a long time ago. We are discussing it now, but I get the impression that we are discussing a policy decision that was made in the past, rather than being a part of formulating policy.
Secondly, I'm broadly worried that we are being too prescriptive in determining what's "hot" and what isn't. I see the HNQ as a way of sowing greater interest in mathematics amongst the SE network, and various members of the wider public. What our more learned members might see as an utterly mundane "low-interest" question, might be still of interest to people for whom the subject never quite clicked (example). For that reason, I can't commit to this idea that we should be constantly fighting the algorithm, which I do think is largely competent at sniffing out general appeal.
I'm almost always in favour of mathematical outreach (though my introverted self finds it personally difficult). Mathematical anxiety and mental blocks are, in my opinion, one of the most serious issues facing our discipline today. The HNQ is a good way of showing people, who don't consider themselves interested in maths, tidbits of maths that they are more likely to find interesting. It's an ongoing debate against all of the common negative perceptions of mathematics (e.g. it's irrelevant, boring, just algorithms, too hard for most people to get, etc).
Basically, I think I would prefer (though this is not a firm opinion yet) for there to be less interference in the HNQ. If a title is bad, for whatever reason, then edit it, and see if the question stays in the HNQ. If that means adding in Latex, then so be it; make sure the question fits into MSE first, and let the HNQ algorithm do its thing.
Primarily, I want to encourage clear questions with well-written answers, and leave questions of interest up to the algorithm. If a question lacks context, then close it for that reason, but I don't think it's necessary to hold HNQs to a higher standard, in the hope that all the context will absorb in the minds of people asking questions to get help with their homework. If for no other reason, I think it'd simply be ineffective. As an example, the question that prompted me to launch this discussion. It provides adequate (but not exemplary) context, in my opinion.
I also don't generally feel comfortable with reasoning based on perceptions about the asker. This includes your disgust at people who title their questions to be clickbait, as well as SimpliFire's comments about people "deserving" of HNQ status. (Maths people being judgemental is another negative perception of mathematics that I prefer to combat, even if there is a grain of truth in it.) Again, I think, edit the title. If they start an edit war, then remove them from the HNQ and maybe talk to them about SE etiquette.
As I said, it's not really a firm opinion yet, but my impression of the policy used so far is that it's too prescriptive and smacks of judgement. I don't really see people "working for the HNQ" as a really big problem, and while context-free questions are a problem, if we keep our usual regimen of closing them (and writing more informative titles), the HNQ should contribute to this problem very little.