I think there is value in this idea.
I've personally had questions closed on other sites, that were then defended and discussed by users I've never spoken to, because they were discussing the value of the question and trying to make it work for their site. It was a pleasure to see people finding the value in my question, and putting effort into bringing it up to scratch.
It was far more welcoming and inviting than being left a comment or close reason that says my question doesn't fit and I need to fix it. As a new user, I don't really have any idea how to make it work for their site, in many cases reading the help section doesn't explain much for my specific question. However, seeing what they did with it, and the discussion around it, actually taught me a lot about what is expected on that site.
I can't recall specifics, but I think I've defended one or two questions here because of what they were asking.
Once a question has been asked on SE, it comes into the public domain and effectively gives the right of ownership to everyone on the site. If we see an interesting question that we like, we should do what we can to make it fit, regardless of who the user is or their attitudes towards the question, site or anything else.
Good content benefits the site far more than it benefits the OP. Additionally, every time we cover a topic properly, we can cross it off the infinite list of potential questions, and future duplicates can be closed and linked to the question that was built by the whole community.
Chat has far too many problems to be used for this purpose. When I used to try and discuss edits I wanted to make to questions, I would usually be interrupting unrelated conversations and got the general feeling that I was being a bit annoying.
At least if we do it here, the only people that will be bothered by it are the people that want to bother coming here to deal with it. In addition to that, anyone can post on meta with no rep requirements. Arguably, it is one of the reasons meta exists. If we made it a formal procedure, we might increase the production of good content. It would also allow people to leave controversial comments on a completely separate post and keep them off the main site, which has been a cause of concern of late.
Overall, I think the benefits outweigh the drawbacks. I can't actually think of any significant drawbacks, except maybe overwhelming meta (not a huge concern imo), but am open to considering any issues I haven't yet pondered.