I'm with Chroma, the advantage of the Driver playbook is similar to the Chopper. First, you have automatic access to something that everyone else has to work or Advance for (gang or car). Then you have moves that make you even better with that resource. A Driver gets a car and No Shit Driver as part of the starting package. That's two advances for anyone else, one for the Move and one for either a workspace with a truck or one of the Driver's vehicle providing moves. In the same way, the Chopper gets a gang and Pack Alpha (available as an advance to some playbooks) but also has Fuckin' Thieves, which requires that Gang to work.
As for letting everyone else drive: sure. I'd certainly let PCs be a member of The Chpper's gang if they wanted, just like they can be in the Operator's crew or the Maestro D's venue. Beyond that, though, they'd need to come by a vehicle somehow (proscriptive or descriptive) and I'd probably give them a bit more trouble than the Driver when it comes to things like fuel, maintenance etc. The fronts mentioned above are a good example.