We should absolutely be closing unsalvageable HNQs faster. Hot Network Questions are the only thing many people see of our site, and I think we should make sure they show off our best. That's why I take a minute every so often to play "hunt the duplicate" when we get a new HNQ. I know it feels awkward to stop someone from repcapping by closing an HNQ, but we really shouldn't let already-asked or bad questions be presented as our latest and greatest. (Relevant MSE proposal.)
I could have sworn there's an MSE proposal to let moderators remove a given question on their site from HNQ, but I can't find it at the moment.
If the question can be salvaged, I see no reason to not save it. If it's interesting - and especially if it has already attracted a great answer - we have the opportunity to polish it up and show it to the world.