IMOIn my opinion, 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 makemakes it to HNQ, 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 guarantee the question gets off the list is closing it, which in the typical case doesn't make any sense.
The HNQ is great especially for telling other SE users about smaller sites they may not know about, but the site should have some way of influencing ifwhether a question makes it or stays onto the list. The HQNHNQ should be for high quality questions that reflect the site and attract new users. As of now, we don't have have any feedback on the quality or that it reflects the site only if it is "hot" or not.
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.