One question in my mind, though, is why is it that that most recruiters solve the coordination issue the other way - first finding companies, and then hunting for candidates to match them. IE, there must be some reason why it's more effective for them to do it that way round, and theoretically this presents an obstacle for you. But perhaps Triplebyte already figured out the answer.
I only know of one established recruitment company which does it the other way round - they don't hunt for candidates - and that's ecm selection. They operate in science-heavy areas mainly, and the chip industry, where there's a particular need for the recruiter to have a good grip on what the role means, otherwise they end up flooding their client with inappropriate candidates. But they are probably not as scalable as you want to be.