I worked a lot of delivery jobs in college and the delivery fee was never for the driver. All the big chain pizza restaurants have some variation of "DELIVERY FEE DOES NOT GO TO THE DRIVER" printed along the side of the box.
The system at a lot of stores tracks order history for each address/phone number, and it was very common for a first time order to never tip, and then begin tipping on their later orders. I'd guess because they read the side of the box, but couldn't really say why.
The only place that ever did pay me the delivery fee was Burger King. When they first introduced delivery, in order to hire drivers quickly they advertised that you could keep the whole $2 fee plus tips. That was such a shit show though, because of how poorly implemented their computer system was and because only about 1 in 10 stores did delivery I would get about 2 deliveries done per hour due to distance. People would complain "I'm across the street from the BK, how the fuck can you still be 'on your way'!" and I'd have to let them know that their BK does not deliver and their burger was coming from the BK downtown. I quit that place so fast to go back to pizza.