A medium of exchange is a facilitator of exchange, A facilitator of exchange could only be purposefully chosen by persons who were already exchanging. Exchange must already be present in order for the medium of exchange to facilitate it. Persons faced with a problem in making their exchanges chose to use a facilitator to help overcome the problem.
The very first trades in history would have been made by the small number of the first human beings who chose to live in community. It isn’t possible that money could have facilitated their trades. If Cain, the farmer, came to Abel, the shepherd, and asked him to accept some of his grain in exchange for a lamb, either this exchange would be made in barter terms or not at all. Money could not possibly facilitate trade until there are a significant number of goods trading among a significant number of people so that some goods are more salable than other goods. In those circumstances, some traders would face the problem of barter and solve it by using a another good to make an indirect trade. This is what money is, a facilitator of exchange.
Historical evidence could never confirm or deny this conjectural history. The archeological evidence of money being used in the oldest ancient cities has no relevance to claim. (Of course, there would be money in the more advanced division of labor in cities, that’s precisely what the theory claims.) And no matter how far back in history an archeological find of human activity may go, something more ancient could have existed for which there is no archeological evidence.