Thank you for informing us about this proposal. Your struggles sound very familiar indeed, and I suppose are to be expected on any site that desires research answers to questions pretty much anyone can ask.
If it could help, I can provide input on where we are right now. After our discussion (over the course of months) we decided on an increased focus on professionals and increasing the amount of expected initial research. For this we introduced a new close reason, 'not framed in cognitive sciences', which (at least for me as a moderator) has made managing suboptimal (unwanted) questions much more straightforward. Although we have no hard data yet, in my opinion the quality of the site has increased since and I have not noticed any pushback yet of people who want it to be more 'inclusive' again.
I would not recommend a new site launch as I imagine you would lose too much valuable content created in the past. To be honest, this sounds like mutiny and is very prone to result in the same problems you encountered in the past. You have a site with content and users (including past users), why not build on top of that? This was our approach, for which I described the process we followed in Rebooting Cognitive Sciences: a Suggested Approach:
That is not to say we did not achieve anything over the past four
years. As a community we have amassed a large amount of quality
content that otherwise would not have been; content contributed by
various users who saw potential for this site but unfortunately might
not have seen it come to fruition. Since we don't have a steady
userbase, we need to find a way to voice the opinion of those that
since might have moved on but were willing to invest in an idea that
they believed in for this site.
In regards to "a site from professionals for professionals" which can "coexist with Health.SE", how do you see this work? Wouldn't the current problems you mention with Health.SE be exacerbated when all professionals migrate to Medical Sciences? Personally, judging from my experience with the 'lack of experts' problem here at CogSci, I do not see this work. This would become Yahoo answers.
Lastly, although some users here might be interested in joining Medical Sciences (or Health for that matter), I see a clear distinction with our site. Therefore, a site merger to me is definitely out of the question (if that was your suggestion?).