Sorry to revive this after a month, but I'm an ego manic and wanted to add my thoughts.
I think Ebok's right. The car is still the driver's 'thing'. While it's true other character's can now get vehicles, perhaps a little too easy, it doesn't take away from the driver as much as you think. It's like guns are the gunlugger's 'thing', yet other characters have easy access to guns.
The driver's simply has a better car, and/or can use it better. Moves like Combat Driver, Daredevil, and just having a high cool stat really let the driver shine when vehicle fights are in play. If anything, having a lot of other vehicles in play just allows you to reflect on how shit hot a driver is behind the wheel.
Sure, the player can choose not to pick these moves, but that's still no different than the gunlugger not choosing AP ammo or NTBFW, then complaining that his assault rifle isn't a whole lot better than the driver's magnum. Sure the tags give the AR more freedom, but even then an AR can be obtained in game. If the player's not choosing them, that's their prerogative. I wouldn't give them for free, but I might push the associated moves a bit so they'll want them next improvement.
For what it's worth, and it maybe too late for nerdwerds's game, I think the fact the diver can choose to plant and still be relevant is more of a sign of improvement and a better playbook from 1e. There's also a lot of driving that can be done, even in a very confined game. An NPC wants someone to drive some sensitive cargo on the outskirts of town to the other side. They've come to the driver because they have a reputation of being a shit hot driver. Plenty of opportunities for road wars without taking the character miles from everyone else.
I also think you should look at the "get with the MC" clause for other character's starting with vehicles carries some weight. The vehicle creating rules say 1-2 strengths and 1-2 weaknesses. It's a free thing, it's totally not being a dick to be stricter on the gunlugger, or outright veto certain choices as part of the deal (the Driver can do whatever she wants though) or trade an obligation or some other bargain for it. Be a fan still applies of course. Even with everyone in roughly equal cars, that's cool. That looks to me like a convoy, which Daredevil specifically mentions. GREAT way to hand the driver some spotlight while having everyone involved in some way.