So turns out the issue is that the radius of the market only indicates the pick up range for goods. To determine if a production building will be supplied with goods you need to check it's individual radius, as shown here:
It's this radius around the production building itself that determines if a production building will be supplied with goods. The radius of the market (or other supply buildings) doesn't actually matter. As you can in the image, the market is just outside the range of the paper mill, so that's why it's not getting any lumber. After destroying the market and rebuilding it like below, my paper mill immediately was supplied with lumber.
Interestingly enough a cart didn't actually transfer the lumber, but it was instead transferred by an NPC carrying lumber. So carts must only be used for the pickup of goods, and the transfer of goods to a production building uses generic NPCs.