Ask HN: Freelance product managers – do they exist?

I've been getting more and more interested product management, spending my own time learning the basics and implementing some of the best practises on my own projects. This led to me being the product manager for an MVP product at my old workplace and for a new charity product (the later being pro bono).

However, now I'm a freelance consultant, I'm not sure if I can get work as a freelance product manager. My impression is that this role is too core to a startup and requires knowing be organisation inside out for the product manager role to go to a freelancer, rather than a full-time employee or long-term contractor.

Do freelance product managers exist? If so, do they work for multiple clients or is it more long-term contracts on one product? Any advice on how to build up clients as a freelance product manager?