No argument that the US didn't have coinage from quite early. The point of the stories is that there wasn't enough of it widely available in many parts of the country until much later, so local "coins" (and printed bills) were fairly common in many places.

Most such "funny money" was not particularly durable, so few samples seem to have survived, but I don't know whether many have ever done significant collecting or that anyone has done much documentation of what they looked like.

Like Confederate money, if the issuing business failed (which was fairly frequent?) they were worthless, so it's likely most were discarded, lost, or in the case of bills used to start the campfire.

Although the stories are common, real evidence is scant; so the explanation is "lore" rather than history and it doesn't matter much whether it's the real truth.

John