IMO, the project-based nature of a flat org is essential because the project is a unifying mission with regular updates and well-understood outcomes (project milestones).
Small orgs are different because motivated people can self-organize. A small theatre company or neighborhood block party committee can self organize because the core cadre of a half dozen people take care of the finances, grants, etc. Even in those cases, someone is at least nominally in charge -- an individual needs to sign checks, apply for grants, hold the cash, etc.
I've lived this at least a dozen times, in a mid/late stage startup, F500 company, and sprawling government bureaucracy. The story is always the same. New business units emerge with a guy in charge and a nimble team of <10 people. They achieve success and grow. Problems start around 10-15 people because folks are focused and cannot productively communicate across the group. Leaders emerge and work out differences. The next point of resistance in around 25-30 people -- the lack of explicit authority undermines the emergent leaders. Everything breaks at 40-50 people unless there is an established chain of command.
There may be a way to tweak these things and make it work, but it takes special skills/knowledge/instincts that aren't obvious to people. Special things are hard to repeat, and business relies on repeatable process.
IMO, there are a couple of ways that flat orgs can work. The most obvious is where the work is delivering unkilled/skilled tasks. A plumbing company has a dispatcher who sends out plumbers. A consulting company has salesmen called Vice Presidents, Project Managers and dispatchers. No hierarchy.
I think the other type of org is a co-op type arrangement. Coops have some hierarchy, but the power structure is elected vs. appointed which changes the dynamic.