TL;DR: My question was this: "Can we have a different set of Q&A guidelines for this proposed topic?" The remainder of this "question" is a lengthy exposition of context and supporting arguments in favor of that idea. But after I gained a better understanding of the Stack Exchange format and community values, I was able to provide a clear and concise answer to my own question below, so I would recommend that you skip directly to that answer. I will leave the original wording of the question as-is. (2014-03-09)
Looking through the example questions that have been closed, and the reasons given, it occurs to me that there is a fundamental difference between the topic of "Mind" and most other topics covered by Stack Exchange sites. When you ask a question about programming, servers or software, cooking, bicycles or Judaism, it's reasonable to expect a fairly broad consensus about the correctness of a given answer. But the "Mind" is a metaphysical entity of emergent complexity, and our best attempts to understand it are still only "soft" sciences, with a great diversity of opinions (and very few concrete facts) about how it actually works. This inherent lack of absolute clarity surrounding the entire topic suggests that MOST of these questions are going to be vague and/or subject to debate. Arbitrary abstract models and statistical analyses of behavioral observations don't readily yield themselves to the kind of black-and-white explanations that seem to be expected by S.E. moderators.
However, in my opinion, this topic is still very important and useful. The general format of S.E. Q&A would still work quite well for weeding out answers that ARE clearly wrong, and collecting a spectrum of plausible interpretations for cross-reference, weighted by consensus agreement. The "validity" of this weighting would be subject to the sample size of voters, community biases, and other factors -- and that's OKAY. Answers and votes don't have to be objectively "correct" to be useful, helpful, or "constructive."
I suggest a reconsideration or relaxation of Q&A criteria and guidelines for a certain subset of topics, particularly this one. Aside from the reasons given above, also consider that some of the questions may come from people who are struggling with mental health issues, and may be having difficulty with rational thinking or communication, and may perceive a rejection in a much darker sense than what a moderator intended. For instance, a vague question about self-worth or suicidal ideation, closed as "Not Constructive", could have a very devastating impact on someone in need of help. At the very least, the specific wording used in the rejection of questions and answers on this topic should be carefully revised to avoid misinterpretations by such individuals, and redirect their inquiries to more appropriate sources of help. That consideration may be relevant to the CogSci site as well.
It is obvious that any Q&A site about the human mind should come with a prominent disclaimer to take EVERY answer with a metaphoric "grain of salt", carefully worded. Along with this disclaimer, any differences in moderation policy could be noted for clarity, to distinguish the site's format from what participants would expect on other S.E. sites.
I do understand that the rules exist for good reasons, and may simply represent a defining characteristic of what exactly a Stack Exchange site is SUPPOSED to be. So, if these rules are set in stone, then perhaps the topic of "Mind" just doesn't fit with the S.E. format. To me, that would seem to be an unnecessary restriction to place on such a brilliant form of organized discussion, but I defer to the opinions of the Stack Exchange community, mods, and admins, with the hope that there is some degree of flexibility when defining an entirely new site.
EDIT: Here's what it comes down to: Self-Help Questions are not accepted on the "Cognitive Science" site, because its emphasis is obviously on Science, and questions of a personal nature are presumed to have no relevance to anyone else. But somehow that rule has been generalized as a precedent to all member sites, even though it is acknowledged that Stack Exchange has no firm policies on the matter. Some comments have alluded to issues of legal liability, but that's what disclaimers are for.
What I challenge you to consider is to allow the existence of a Stack Exchange site that welcomes personal questions, and allows more open-ended discussions, and even some debate, as long as the arguments are grounded in credible references, with sensible restrictions against giving specific medical advice. Even if a particular question seems unique to an individual, there are surely other human beings out there who can relate, or find the discussion valuable in some way.
I know that some of these questions CANNOT be answered exactly as they're asked; in particular, if someone gives specific treatment recommendations or advice about medications, those types of answers should be removed by moderators. When questions solicit that kind of advice, an acceptable answer would politely define the limitations of information available through the site, and give helpful suggestions about how to seek professional help. Exactly where to draw that line may require some clarification, but most of it could be boiled down to a few template responses, which could be given more tactfully in the form of an answer, rather than "CLOSED: Not Constructive."
The argument could be made that duplicate questions would abound, to which I'd reply, "So what?" When a question is "closed" it isn't deleted -- I've found "closed" questions directly from Google SERPs. I don't see any particular advantage in closing a question, versus providing a polite referral to professional help in an answer, and letting that answer compete for votes along with other answers that may attempt to provide more specific help -- trusting the community to vote appropriately.