I do agree that it fits the theme and setting of the game for the player to be put into (seemingly) impossible situations. I was never angry or frustrated about failing the delivery but I knew with certain destinations I was likely to fail. Setting the timer to 30 when the pizza is picked up is, I feel, the more 'gamey' approach, I'd support your decision either way!
I also wanted, and forgot, to mention that I appreciated the car not exploding whenever I slammed it into a wall or building and giving me a massive repair bill or ticket for public property damage. :D