The problem is the billing cycle, not the paperwork
A paper POD travels back with the driver. Until it arrives, the trip cannot be invoiced, which means the delay between delivering goods and being able to ask for money is the length of the return leg plus however long the document sits in a bag.
For a single trip that is an inconvenience. Across a fleet it is working capital sitting in vehicles.
The second cost is disputes. A consignee querying quantity or condition weeks later is arguing against a signature on a crumpled sheet, if it can be found at all. See the paper POD racket.