First of all I want to thank you for your engagement for doing changes at HMPs. The features added seem good and helpful.
"Are there any pain points that currently exist with this feature (even if it is only on site specific) that you would like to see addressed?"
One big pain point for me is that there are only such less posts addressed as HMP in the Community Bulletin (facing specially Stack Overflow but might be elsewhere, too).
I think there can only be max. 2 or 3 posts marked as HMP simultaneously which stay there then for about 1 or 2 weeks (my guess), which is IMHO problematic. Many posts which deserve being marked as HMP will get lost on the Meta site and many user who are not confirm about Meta or are just not so interested in Meta in general have no ability to share their opinion on these posts, which is bad.
This is also because currently to become "Hot", a post need to be not older than 2 weeks, but if other question take the rare position, there is no ability to give them the publicity they deserve.
Speaking about restricted age for becoming "Hot" there also should be an improvement made.
I do not say I want hundreds of HMPs but just a bit more (best would be 10, fine would be at least 5 at the same time).
If the HMPs section in the Community Bulletin has too less space available, maybe think about to implement a special page where all HMPs are listed.
Or another solution as mentioned by @Trilarion in the comments would be to have a list of HMPs and just show 3 of them, cycling through the list each time the page is refreshed.
Any of such an improvement on that point would be a big improvement in my eyes.