I'm working at Papa John's and we don't charge for delivery, nor is it built into the cost of the pizza, I don't think. Right now, you can get two large two-toppings for $16.80 (for example). This is the price whether you walk-in, call ahead and pick it up, or have it delivered. Unless they're charging the walk-in and pick-up customers for delivery, Papa John's delivery is free, just as advertised. My $.02, anyway.
And to answer the question posed, yes, I would probably pay the extra buck or whatever for delivery. When I order pizza and have it delivered, I'm on campus at school and my car is not convenient enough to go walk and get it, pick up the pizza, take it back to the room, and then park the car. When I'm at home and we order pizza, I just go pick it up.