Oh boy.. I'm fully aware that I'm opening a family-sized can of worms here but I believe they should be allowed in a controlled manner.
From how I read it much of Monica's reasoning stems from critique-based questions not really fitting the original Stack Exchange model of there being a broader benefit beyond the original poster of the question, and I'm just not sure that's true.
Monica likened offering critiques to Writing SE becoming a "writer's circle" and that's actually where I'd say it can offer benefit to others.
Certainly from my own experience in writing groups both social and those and on creative writing courses in an academic setting I've found that the critiques of other's work to be valuable to my own writing. Just because I might not have written (or ever write) the same passage being critiqued doesn't mean I can't apply the same good techniques and practices to my own.
I think there's a good analog between a critique request for writing here and the existence of whole Stack that does the same for programming and they've managed to make it work, and work well.
Yes, I think implementing it here would bring challenges, and that's why I mentioned that they needed to be controlled. So what would that look like?
Length - I think a reasonable maximum length needs setting, you can't expect people to wade through pages upon pages of someone's writing and provide a detailed critique on an SE site. For that sort of thing the existing Beta Readers chatroom is a far superior solution. Obviously it needs to be long enough to be worthwhile but I think 200-250 words as an upper limit would be sufficient, even if it weren't impractical for a whole host of reasons posting whole works on SE would potentially run into issues with users unhappy that they've just posted the whole of their novel as Creative Commons!
Concrete question/goal to address - I think at least the notion of a single "question" or problem needs to remain, and for that not to be too broad and not too opinion or personal taste-based (just as we require questions to be now) - "Is my writing any good?" wouldn't work clearly, but "How can I increase the pacing in this scene?" or "I'm trying to show this character's sense of building anger here, does it or if not how should I change it?" would be more like it in my opinion
Answers - there would also need to be the other side of the coin, the answers would need to provide good detailed critique. "I liked it", "This sucks" aren't good answers, the why of an answer would be as important as ever, and should relate to the writing not opinions, and the how to approach improving the writing is critical. Without those things it's not an answer - it's just an opinion.
There's still potential pitfalls though - for a start a max length for the excerpt is something that wouldn't be controlled by the SE system, it would rely on high-rep users and moderators to actually check and handle accordingly. Which is a headache - one you can mitigate by giving an approximate range and only acting on those who obviously or egregiously exceed it, but still it's inevitably going to result in someone getting annoyed when they have a question closed for length reasons.
I'm sure there's one or more people reading this and thinking "But moto you silly duck, what your proposing is just the same as using an example of your writing to illustrate the problem/question and that's already allowed!", and you'd have a point, partially, but I think what I'm proposing is that we move the writing front-and-center and allow somewhat open-ended feedback even if it strays from the specific question - more like how Code Review does it. As it stands anything that has even a whiff of being a critique-request to it is off-topic.
What To Write
This I'd firmly say should remain off-topic, there's much better ways to do similar things than asking a question on a Stack Exchange site, better ways within the Writing SE community even - chat, the writing prompts/challenges on meta etc. This isn't to say that answers can't include plot advice or suggestions as part of answering an on-topic question but we aren't a suggestion box for people struggling for inspiration and that frankly that would just descend into a mire of rubbish content very quickly.
I think we already strike a good balance here - "straight" grammar questions have their home on EL&U or ELL SE already, and we damn sure ain't a spellchecker. So I wouldn't change the way we currently handle this.