No need to hedge-- meta is the best place for meta questions!
We can't.
1. What does it mean for an answer to be "correct" here?
As pointed out in comments on the question, "correct" is kind of fuzzy for what IPS deals with. Even trying to assess based on answers that "work" is hard, because there is no real way of knowing how many alternative approaches would "work", and (very importantly) there are nuances to situations that we simply can't observe here.
Two questions might be on the same topic, have the same basic setup, and ask for the same outcome, but the real, fully-defined states of those situations might not be identical. Those differences could easily be enough that an answer would "work" in one but not the other.
As a hypothetical example of this, consider questions about saying something difficult to someone without that person getting angry-- that's going to depend enormously on the person being told something, and we're simply not going to reliably get a good picture of that from questions.
2. What do upvotes and downvotes indicate?
I think that on the whole the voting system is good for identifying answers that the community feels have value. There is a lot of ancillary information that gets conveyed this way (though with poor resolution).
However, even on sites with falsifiable topics I think that it is pretty unlikely that every voter is actually testing the solutions offered and voting accordingly. Obviously, some do, but I'd bet that an answer with 100+ upvotes has a lot of "this looks like it would probably work" votes. I've seen questions with highly upvoted answers which are simply, demonstrably incorrect-- falsifiability isn't an absolute defense.
And downvotes can come from information totally separate from whether or not the posted answer is correct (however we want to define that). A correct answer might be downvoted because a user felt it had the wrong tone, for example.
Votes are simply not a high-integrity representation of specific information. They're more like Reddit upvotes or Facebook likes than formal sign-offs on code reviews. Whether or not that issue is worse on IPS than elsewhere (and I suspect that it is), I think that you may be over-valuing the information content of vote totals here.
3. What might be better?
I want to say upfront that I definitely believe that improvements are possible (including in ways I've not yet imagined). But I feel that the improvements are far more likely to be marginal than revolutionary.
We're always trying to strike a balance between site usability and reliability of answers. The "back it up" policy is a good example of this: we prefer answers to contain rationales for why the answerer thinks they will work, whether that's a sequence of logical reasoning or drawing from personal experience. We generally can't verify that someone who claims to be answering from experience is being honest, but I'm not sure that that's a solve-able problem here.
As for referencing research, I am on record as aggressively opposing that idea. It would be nightmarish to moderate, difficult for site users to do (are you going to read a dozen 50-page research reports, at $30 apiece, before voting on or accepting an answer?), very frequently isn't more conclusive than what we've already got (studies are almost always pretty narrowly constructed in anything that would touch on IPS), and the peer-reviewed publication system is designed for ongoing discussion among certified experts (and not black-and-white Q&A for specific instances, which is what IPS.SE is).
In a lot of ways, I think that relying on citing publications for answers inculcates a lot of these same problems and then hides them under a veneer of scientific validity.
As examples, consider the Milgram experiment and the Stanford prison experiment. People held these up for a long time as indicating a definitive frame for how humans behave. But over the last decade evidence has come out that there were some pretty serious methodological problems with those studies.
Is a standard IPS user equipped to evaluate how meaningful those problems are while focusing on the content of a question for which those studies are only background information? Do they still count as evidence in an answer? Would an answer based on a theory of human behavior derived from those studies be better or worse for citing those studies explicitly? Is an answer that misrepresents or overinterprets the results of a study better than another answer which cites nothing at all?
As much as I would like a better and more objective way to evaluate the answers here, I'm not sure that it's realistic. It's a very subjective domain, and questions are already heavily influenced by factors which are invisible to all involved, and then interpreted by potential answerers subject to another largely unobservable set of factors.
