Questions on the "Hot Network" list are like travel billboards: they advertise sites to users elsewhere, to maybe get those users interested in participating. But I have noticed that those questions don't seem to get edited any more than others; bad titles pop up in the list, and they stay. I studied this over the course of a few days, and I collected a fair number of samples.
This makes me sad. One of the great things about this platform is the ease with which experienced users can improve others' posts. These posts have interactions from those experienced users, but they're left looking a bit shabby.
I'd like to propose an automatic banner be placed on questions that are featured on the HNQ list. The banner would be visible to anyone with the editing privilege, and would say something to the tune of
This post is on the Hot Network Questions list! That means lots of people are seeing its title and deciding whether to come visit the page. Is the title grammatical and descriptive? Is there anything else that can be [edited] here? Help make your site look good for visitors!
It might also be worthwhile to give experienced answers a little pop-up after they post an answer
Thanks for answering! Please take a minute to look at the question and  if necessary to make it as good as it can be.
Here's a few examples. I've noted the title's problems and taken a picture of the HNQ entry. I visited the questions as I saw them in the list and looked at the circumstances: how much attention the question had gotten and from whom, and how old it was. (Images link to the questions if you want to visit them in their current state.)
While I was collecting these, I entirely ignored sites with deliberately "weird" titles: Arqade and Puzzling, and to a lesser extent Code Golf. This is just about objectively bad writing: mostly grammatical or spelling problems, and also titles that don't provide much in the way of information.
Five answers, two by top users. No edits (and the body also needs grammatical help).
Edited by a high-rep user, title untouched; well-voted answer.
Then there's the really low-hanging fruit. This stuff is so easy to fix up; it's almost mechanical.
Now, this isn't a huge problem by any means. While I was actively looking for these things, I never saw more than three awful titles in a list of approximately twenty. At the absolute maximum, 5% of Hot Network Questions have titles like this.
On the other hand, I think it could have an important impact on individual users' attitudes towards their sites, opening their viewpoints a little to consider what the place looks like to inbound visitors. And if more improvement does happen, it will only make the network as a whole look better.