I really like the HNQ, and have often learned about interesting topics because of it. I would hate to see it go, and I don't think it actually has serious problems. The fact that SO felt the need to remove some items from it after a random person on Twitter complained about it seems like an overreaction to me.
(Those IPS questions looked genuine, asked in good faith, on-topic for the site and correctly titled. I hope you're not going to censor contentious subjects; one of the most interesting questions I saw in the HNQ recently was titled "My 3 year old thinks she's white; should I correct her?")
Having said that, there is one thing about the HNQ that annoys me immensely: the fact that I can't choose to hide certain sites. I have no interest whatsoever in Gandalf, Dumbledore or R2D2, or what type of stone victims of the Petrification spell turn into, or whether it is a sin to ride a unicycle on Tuesdays in a certain religion.
The fact that a lot of the tiny site icons are similar and hard to recognize means I'm often halfway through reading a title in the HNQ when I realize it's from one of the nonsensical sites.
So please, let me click somewhere so I never have to see Role-Playing Games, Science Fiction & Fantasy, Christianity, Buddhism, ... ever again. (But don't group them into categories, because I quite enjoy World Building, e.g.)
(Of course you don't have to agree with my selection of uninteresting sites, because that is part of the point I'm making: everyone will have adifferenta different selection.)