Lord Rimbaud (OK, so I'm guessing, but I'll stick my flag in the sand on Lord as opposed to Lady (blush))
I'll try to comment ('answer' would imply I think I, like, know stuff and should be listened to. Heck, even my _cat_ doesn;t _listen_ to me! (blushes again) in order of your questions. If I mess up, I'll at least try to be vaguely coherent. I'll probably mess up, but hey, my wife tells me I'm trying, so it's worth a shot!
"This presents two problems for me:"
OK - let's take a run at them sir :-).
"A) I have already written the piece"
That's not actually a problem. Whether you've written the piece/ book or not, I think the approach is the same. For non-fiction you:
Identify your market
Identify which Publishers/ Agents (generally non-fiction goes more often direct to Publishers than fiction does) play in that playground
Ask them if they'd be interested (the Proposal). Again, it doesn't matter if the book is written or not, any more than it matters if a fiction writer writes a book she/ he doesn't intend to market, but changes their mind later. Apart from anything else, the very act of writing is one of the best training plans there is.
"B) By most standards I would not be considered qualified to render judgments or ideas on various phenomena, whether it pertain to scholarship or aesthetics or politics etc"
This is where it gets, to me at least, both more interesting and harder. Because, whether you write the most amazing books on earth may not matter. Yes, I really said that. Or rather I didn't - someone I wish I could remember, could find the article, did. It was an Agent, talking about fiction (but I think her point applies equally to non-fiction). I can only remember she was female, but she said that when she talked to authors, a lot of them thought her biggest problem must be the bad writing she was sent every day. Which, she said,wasn't true. A lot of what she was sent was _great_ writing. But The Next Great Work of Literature wasn't what she needed, wasn't what she was looking for. Because if she was going to invest a year, two years of her life in a book, she didn't need Great Literature, she needed dollars and cents to put food on her family's table. She needed Great Sales not, at least not necessarily, Great Literature. And, she said, so did Publishers. _They_ wanted bread on the table as well. So what she would look for was reasons a submission would break sales records.
I think that's equally true for non-fiction.
So what makes a book sell? Well, if I knew that, I'd be rich. But a number of things can help. A new book by me on Super-String Theory wouldn't attract as much attention as exactly the same book by Stephen Hawking. Why not? Well, most folks in the field know Mr Hawking knows what he's talking about. Me? Not so much, Super String Theory or Knitting String Theory. So being respected in a field can help a lot for non-fiction. But here's a thing - if (at the risk of being sued for taking her name in vain) Kim Kardashian released a new book about diets, it would sell. Oh, sorry. I mean, it would SELL. Whether or not she's a qualified dietician and nutritionist. It would _SELL_. Why? Because she's famous ,already has a defined audience. If you can show a Publisher you have one of those, you're a long way forward (even if you're not a Kadashian). A viral video of a dog who likes your guitar music, hates it when you stop?
That's a hair short of fourteen MILLION views right there. That's an audience. That's a hook.
But we can't all play the guitar, never mind dog-a-liciously. But we might have a web page, a social media site - 'Fifty Shades of Grey' was, at least in part, picked up because the writer had a bunch of folks on a site who were avidly reading her stuff.
So, cutting an already long set of words short, one of the questions you have to be able to answer for a Publisher, especially with a Proposal, is - 'why would people buy this book? Read this book?' It doesn't matter what the answer is - but having one you can show a Publisher helps.
"Are you, Smith, saying that one whose proposal is accepted WOULD GET PAID while writing the piece"
Er - nope. Sorry :-(. Or not generally. Or probably not. Or maybe. What you get is a form of agreement that says 'sure, go write it. Or send it if you've already got it. But send it direct to me, and if it's good enough, we'll put it out there - _then_ pay you $Thismuch.' If you're really well known, or the best in your field, you _may_ get paid for the writing exercise itself, but not generally.
"Am I right about this? For my non fiction stuff,
WOULD IT BE WORTHWHILE to send a proposal in which I say, a) I am not asking for a dime while I am writing this as it is already written and b) perhaps you will be interested in this even though my qualifications may seem dubious because I will not be paid prior to the time you accept this."
It depends. On what? On how you answer the question they won;t ask you up front, because they'll expect you to answer it without being asked. To answer, in a way they believe, why people would pick _your_ book off the shelf, hand over their dollars, _Read_ it - and tell their friends to do the same. Whether that's fame, dog videos, expertise in the field, a series of articles on web sites where people actually start discussing your thoughts in the comments, and in large numbers - true, false or marketing, you need an answer.
So how does that feel as a start? Not just to Lord RImbaud, but to others here as well, who likely know a thousand times more about the field than I do? Over to y'all... :-)