Do you think that setting up similar a room as on Academia - with a feed of the HNQs from Mathematics Stack Exchange - could be useful for some purposes?

One potential use would be that this would be place where it would be easy to notice which question entered the HNQ list. (And thus HNQs where some action is needed would not be overlooked.)

Looking at the linked room, it has rather low number of non-feed messages. So I guess not much discussion is going on in that room - it would be probably similar here.

But still I can imagine occasional short discussion whether some question should/shouldn't be removed from the HNQ.

And perhaps this would be also room where some interesting questions would be listed - if we take an optimistic view that interesting content has slightly bigger likelihood to enter HNQ than an average post.