I don't believe it would affect the delivery, just the "tracking". Not sure that USPS uses the tracking for routing purposes.
Shippers don't make up numbers, they are created or indexed by USPS via a database, and assigned to each package. Your system has to be online/connected to the internet to create a shipment and receive a tracking #.
DHL didn't. I wasn't involved in that, I did maintenance/repairs on equipment but routing and tracking (for the customer) were seperate but on the same label. This was 10 years ago though. If an employee failed to scan a package anywhere the tracking would be broken but the package would still get where it was going unless someone put it on the wrong conveyor. Then it'd get there with a bit of delay.
Presumably that's what happened with the USPS package, they put it on the wrong conveyor belt and it headed to Puerto Rico.