IMO, this proposal is treating the symptom rather than the cause of the issue.  The sites themselves pretty much don't have any input on whether  or not a question make since especially on smaller sites a few quick answers and votes can quickly put the question on the HNQ regardless of quality and how well it fits the site. Once it's there, the only thing the site can do to gaurntee the question gets off the list is closing it which in the typically case doesn't make any sense. 

The HNQ is great espically for telling other SE users about smaller sites they may not know about, but the site should have some way of influencing if a question makes it or stays on the list. 


A simple suggestion is once a question reaches a certain threshold instead of just going directly to the HNQ bar, it is instead put into a queue to check the quality of the questions similar to the triage. Based on its score there the question will get a multiplier to its HNQ score which won't influence it's overall position on the HNQ, but determine if the site wants it there over other questions that have similar scores.