What would you think of a system that instead tracked materials and manufacturing process?

So, each step in the manufacturing process would use proofs received from previous steps that the amount they claim to have produced is not more than they could make with only the materials that previous steps have said they supplied?

So, e.g., if there is a place that makes good(?) grain, they would include some cryptographic signature that they provided the amount of grain, and then a baker receiving it would be able to produce a signed claim that they made so many breads with only grain from there, and they wouldn't be able to fake making more by adding grain from elsewhere, because the total of all their claims of how much of the grain is used in each bread could not total more than the grain which was signed as provided?

Or something like that.

I don't remember the specifics.

Would it be better to track who owned the money before you, or to track how the product was made? Or both?

I think there are probably some advantages of anonymous money, but I don't know that they outweigh the costs that you mention. (I suspect non-anonymous money makes auctions and things more difficult, and might reduce effeciency, but that might just be the acceptable cost of influencing the world against people doing harm)