Carl Menger's barter story is one of the most successful myths in economics.
The problem is not that it is theoretically possible. The problem is that there is little historical or anthropological evidence that societies actually developed money this way.
As David Graeber pointed out, anthropologists spent more than a century searching for examples of economies organized around pure barter and repeatedly came up empty. What they found instead were systems of credit, debt, obligation, and social accounting. People kept track of who owed what long before coins or commodity money became widespread.
The familiar story of a blacksmith searching for a farmer who wants horseshoes is largely a thought experiment. Real communities were not collections of strangers conducting spot transactions. They were networks of ongoing social relationships where obligations could be recorded and settled over time.
Historically, money often emerged alongside institutions capable of measuring and enforcing debts. Temples, palaces, kingdoms, and states maintained accounts, levied taxes, and denominated obligations in units of account long before everyday markets were dominated by coinage.
Coinage itself was frequently linked to states paying soldiers and then demanding taxes in the same currency. This created demand for money not because markets spontaneously selected it, but because political authorities structured the monetary system around it.
The historical record suggests that credit came first, coinage came later, and barter usually appears at the margins when monetary systems break down, not at the beginning of economic history.
The real myth is not that states played a role in the development of money. The real myth is that money emerged from a world of isolated traders solving the double coincidence of wants problem through spontaneous market evolution alone.