A gta style driving wouldbe fun, maybe give you the option of one or two vehicles per playthrough. Have like an atv, motorcycle, dirt bike, corvega, pickup, road warrior esque car, Brahmin cart, horse, gyro. A few more.
As for running things over, depends. With the atv and the bikes you can run down humans and smaller creatures, but risk being knocked off. Don't even try going for a deathclaw or giant scorpion, that's suicide. You can also shoot small arms whilst driving and use some melee weapons to attack. Maybe you can mount a heavy weapon while stationary. Carries very little
The gyro can be used to reach unreachable places, and to get a beautiful look at the landscape. Creatures cant touch you whilst in flight, but bullets can. You are vulnerable while starting up the vehicle, but the rotor can defend you a little bit. This vehicle is the weakest. Carries the least
The corvega is average in terms of speed and power, at max speed you can mow down some of fallouts bigger creatures causing leg injuries. And it will obliterate humans. It carries a nice amount of loot in the trunk and in the backseat, you can also have your two companions in it. Works best on roads but can go offroad at the expense of tire condition. You can do drivebys and have a companion Fire big weapons.
The pick up is by far the rugged mans choice, carrying many pounds of equipment and it can even be used to carry your bikes or gyro. It can decimate most living things. It is great on and off road, however it's a gas guzzler and costs a lot to maintain. you can set up a companion in the back to shoot shit.
The brahmin cart is average, can go on and off road. The brahmin can attack but are weak, best used for travel only. You can fire weapons in it
The horse is fast and goes off and on road you can use melee weapons on it, it has a special effect where some creatures are scared of it
The road warrior esque car is similar to the corvega, but only has 1 seat and carries less. Can run down smaller creatures.
The cars can be flipped and can cause more harm than good on you, but they are also quite efficient in terms of gas, item carrying and companion compatibility.
If your gyro is shot down, the only way you will survive is landing in water or if you can land on flat ground. The horse can be killed so can the Brahmin. The truck can break down easily.
If your cars break down you have to leave it and find the parts needed, bikes can be dragged to safety as can the gyro. If your animals die you can have a perk similar to puppies, or you can buy a new one.