People drive down the road to get to where they're going. They're probably in every day type cars. IN the apocalypse world, maybe those cars are diesel fueled war-machines, but the in the end, they're still just the trash people put together.
In my games, the driver is the luxury car, the racer, the real damn tank. For whatever reason they've got a relationship with their vehicle that tends to go their way. The vehicle and the driver both care about each other, and it shows. The other crap on the roads just don't compare in the same way, hell, the other cars are probably jealous.
Driving is a drivers art.
The driver also has access to vehicles that other people just don't. If you've got a bunch of people sailing on the coastline with rickety old fishing vessels loaded with arms and small speed boats with front gun turret. The Drivers the guy I'll let say. I want a Helicopter, a Submarine, A hovercraft for the marshland, a small airplane. They get access to things that change the scope of the game.
Drivers don't escape threats. He can try, but the threats might still come after them, and word-gets-around. The driver also generates threats, because he's got shit that other people want. He can access things that other people cant. When you've got a driver on the move, then probably the most relevant threats are also on the move. The game probably covers a larger area. Just because the driver moves, doesn't mean he doesn't need the road. Or he doesn't need friends along the way that help him resupply. Think Firefly (the movie) and what happens to Shepard with the Crew tries to go to ground. A driver still has to eat.
I dont do mad max type of games, not even close. But My games normally encompass a larger area. Usually it all plays around a few hardholders and whatever is in between. Getting through the middle of that and all the threats, that's always dangerous and I'm always pushing problems in into it. The driver can get ~anywhere~ more reliably then someone else, but that doesnt mean he's just gonna drive off into the sunset.
Drivers also pretty much swallowed up the operator. Far as I'm concerned, all the cool things an operator could do are now in the drivers hands. So I talk about how he gets his jingle, and play into that pretty hard. These places probably sit pretty still, and thus are vulnerable to all the things the driver might out run.
As for playtime, keeping people on scene can always be a challenge. Doesn't have anything to do with a driver though. I don't see how its related. I rarely if ever have all the players in the same place at the same time. That's just too much firepower, and they're not always friendly towards each other.