I'll talk a bit about why I think this is so difficult, though I don't mean to say that this is a reason to not attempt to make the site more professional, just why I think it hasn't become that way, because I think it's important to think about these things if you want to make something change.
Neuroscience/psychology is very big...this site is very small
I don't have much experience on all the other sites you mention, but here we are very small. I do have a little experience at Cross-Validated and my experience is that nearly all the questions there still come from students, with the majority of those coming from lower-level students. Occasionally some research-level questions come up, but that community is just so much larger than this one and still most of the questions are very basic.
I respect a lot of the other contributors here, but frankly, I'm not qualified to answer questions in their neck of the woods and they aren't qualified to answer questions in mine, at least not better than I can answer them myself.
Answers at the forefront of neuroscience rarely fit the SE mold
Even if experts exist here on a certain topic, the correct way to summarize the state of research in a field is by writing a review article. I'm hesitant to ask advanced questions, especially in unsettled fields, because I know the level of effort an answer takes is not appropriate for this site.
I often find the more advanced questions people ask here fit into this category. They are looking for the "real answer" or the "definitive state of the art ideas on ______" - these questions tend to go unanswered not because people here don't know how to answer them, but because they aren't going to spend hours explaining why there is not agreement and the range of existing evidence. I find they are often asked by people outside of neuroscience, approaching from a machine learning/computer science/mathematics background, looking for an easy answer by which they can avoid learning biology from the ground up.
I have tons of questions about research in my field. The way I get answers from other people on these questions is through things like a journal club, where a bunch of people get together and dive into a paper or group of papers. Unless we decide we hate a paper and therefore the summary ends up being "this paper sucks," there is no way to distill that hour-or-two discussion into a SE answer. SE is not designed for the type of back-and-forth discussion that I find actually leads to better understanding in those contexts.
Experts are good at finding their own answers
By the time someone is able to ask research-level questions, they have a lot more experience in finding research-level answers. If someone here has a research-level question, they are definitely more motivated to find an answer on their own than anyone else here would be, more familiar with the precise type of answer they need, and also likely to be interested in any auxiliary information they come to along the way.
That doesn't stop many of the experts here from asking good questions, but those good questions tend to be asked outside their specific field of expertise.
As an exception, I love the example that AliceD linked in his answer: good question, good answer...and the answerer inadvertently used as support a reference (patent) on which the question-asker was an author.
