The biggest problem is that the business owners want management to represent their interests; if they were elected by the workers, they would more naturally represent the interests of the workers. It might work out in a worker's co-op where the workers and owners are the same people.
I'm trying to understand how a "flat org" like this is different from a co-op. Seems like the "flatland" companies give you autonomy without ownership, which seems bad to me. It's only flat so far as the actual owner allows it to be, and you end up with a bunch of people with autonomy with no stake in the outcomes.
Co-ops still have boards and often employees that report to the board (or another employee). There's a structure there, but they also tend to be smaller so the structure has less impact.
I was treasurer of my housing co-op and we had one full-time employee who reported to me and the board. That employee (the coordinator) hired contractors for maintenance, bookkeeping, as well as things like the laundry machines.
Even large co-ops (like clothing stores) have the members vote a board in that sets up the structure.