Why couldn't there be a history where people were keeping track of debts and then started trading the things on which they kept track of debts and then whammo! You got money.

There can be and it would fit my understanding of the theorem because there's a connection between the new money thing and the previously valued debt thing and a connection between the debts and whatever valuable thing they came from.

I'm pretty sure your last paragraph fits the Austrian model. The point would be that the tradable thing can't be arbitrary. It has to be based on what people already value.