In a for-profit context, worker cooperatives are firms that are owned and managed by workers. Pérotin (2012, 2015) summarizes research to show that worker cooperatives have positive impacts on both firm productivity and employee welfare; there is a lot more research showing that worker ownership is modestly better than regular capitalist ownership but I won't get into that here.
There are plausible reasons to explain why the private sector would generally refrain from adopting worker cooperatives even if worker cooperatives are better: owners and managers have a self-interest to keep capital and decision power to themselves, and people may have a pro-capitalist bias. So the rarity of this idea is not good evidence against its correctness. (Mild forms of employee ownership are fairly common, however.)
Nonprofits can't be owned, so a cooperative nonprofit would just be about worker management (i.e. democracy). This is called a Worker Self-Directed Nonprofit. I didn't find any research showing how well this idea works, though of course it seems good based on the results of for-profit worker cooperatives. If you search the phrase Worker Self-Directed Nonprofit there are various sources of (generally cheery) commentary and guidance. I'm not sure if it's been tried very much.
Workplace democracies, and WSDNs by extension, also seem to have a generally positive moral PR tint. Though a minority of people may perceive them as silly and inefficient.
Overall, experimenting with WSDN in the EA community seems like a valuable idea.
I haven't worked for an EA organization, I just wanted to throw this idea out there.
My experience with organizational design is that the formal structure tends to follow not lead the informal structures that arise among the people in the organizations. Yes, over time organizations become "ossified" such that the formal structure also creates the informal structure, but this is not much the case in early and small orgs, although there are usually some exceptions to this as certain formal relationships develop early, such as the founder(s) or some other persons having authority via legal and financial control that backs their ability to influence others and hence seeds the creation of the org structure.
Overall this is to say my guess is these sorts of structures are either already naturally arising and where they don't it's because there are other incentives that push those organizations in other directions.
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That's one way to explain my thinking. Another is this:
I read your post as suggesting something like "hey, what if we tried this different org structure; I think it might be better", but to actually try a different org structure you have to have people who want to relate to each other in a different way. It's typically only at large orgs with ossified structures where people are not relating to each other in the way they would like and where suggesting a change of org structure might manage to shift an equilibrium by getting everyone to re-coordinate towards something they prefer.
In a small org you probably can't make the structure much other than what it is unless you first change the people who are creating the structure to be the kind of people who would create the desired structure. That's because I expect the existing structure to already be a natural equilibrium that is roughly correlated with the kind of structure desired proportional to the amount of (official) control each person in the org has. Thus unlike in a large org there is not a hope that you can hit reset and get a different outcome by breaking the existing inadequate equilibrium.