Hmm, are markets self-ordering?

I think in the purest sense, just having some form of common currency would be considered "order" - unless you're suggesting that only pure barter is necessary for a functional society - but then even that requires some broad agreement on weights & measures in order to function (e.g. how long is a board-foot? how much flour is 1 kilogram?). Collectively the market must agree on some standards in order to function.

Also, unregulated markets always end in monopolies, with or without a capitalist economy. Sooner or later someone will try to establish themselves as the sole supplier of some good or service. Only active oversight and regulatory enforcement can prevent it.

Yes, I always form my socioeconomic conclusions based on television sitcoms, they're the best source of opinions.

Sarcasm aside, the idea presented here doesn't work for any more complex form of cooperative labor (e.g. public water systems, telecommunications infrastructure, hospital services, large research projects, bridge construction, etc). Someone has to perform administrative duties to organize the work being done and ensure that the needed materials and experienced personnel show up at the right places at the right times.

Like, try telling the head surgeon of an OR that the staff can be left to figure things out for themselves, and all the operations will get done correctly as scheduled without killing any patients. They'd laugh you out of the building. The surgery schedule would be chaos by the end of a single shift.