Quoting relevant pieces from the OP's question, emphasis mine:

> All I'm attempting to say is that with the right infrastructure, subjective questions do have a useful place in the exchange of ideas. In a way, such a site would be similar to Wikipedia; but it's content would be *organized by popular vote*.
>
> I know this is no little task, but is there value in doing so? I think there is. *Nothing is going to prevent people from asking ill-formed questions. Wouldn't it be nice to somehow accommodate these in an intelligent manner?* In this way both objective and subjective questions would have their own place. And people that like Stack Exchange the way it is would actually benefit. Lots of flagged-and-closed questions remain on this site. Such questions could be migrated to the sister site so that only the purely objective ones remained here.

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Responding to individual points:

> [Stack Exchange subjective] content would be *organized by popular vote*.

I think Grace Note did a good job explaining the issues with handling subjective questions on Stack Exchange in [this Arqade Meta post](https://gaming.meta.stackexchange.com/questions/997/handling-game-recommendations-how-can-we-solve-these-two-problems-of-quality); see the discussion about Repositories.

The problem with voting for answers to subjective questions is that the votes indicate popularity instead of suitability... that's why these discussions are subjective.  For instance, suppose I ask this question: "What graphical design program should I use?"... 

Do I have to mention all the possible answers and religious battles about Gimp vs Adobe Illustrator vs Inkscape vs Visio...  Of course, the problem is a vague question, and voting for answers to vague questions tells *me* very little about solving my problem... What do the votes really mean other than *"Other people like this software for what they think the (extremely vague) question meant"*.

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> I understand that Stack Exchange is not built to support open-ended questions and discussions...

Actually, that's only partially true.  Stack Exchange's main site doesn't support discussions, but we do have chat.

> I know this is no little task, but is there value in doing so? I think there is. Nothing is going to prevent people from asking ill-formed questions. Wouldn't it be nice to somehow accommodate these in an intelligent manner?

We have a pretty successful platform for well-defined questions; Stack Exchange has built a [monumental reputation in the span of a few years](http://www.alexa.com/topsites/category/Top/Reference) (currently ranked #3 in Reference by Alexa) because we do focused Q&A so well.  I'm not sure it's worth diluting our success to target something (subjective discussion) that's covered well by [reddit](http://reddit.com/), and [yahoo answers](http://answers.yahoo.com/).

We are known for Q&A... if someone isn't sure what their question is, or it's too subjective for the main site, they can still ask a lot of subject matter experts in [Chat](http://chat.stackexchange.com/) assuming the have 20 points on the site.

It's not a perfect solution, and it doesn't have answer votes; however, I don't really think voting helps subjective questions.  Subjective questions need discussion... chat is the best we have.