A downed power line might well still be live, especially the lower voltage lines.
If a cable drops onto your car, your tires should insulate you from any electric shock risk by preventing the current from running through your car to ground.
Even if you have no tires and are running on the rims, the metal body of the car should route any current around you rather than through you. The power guys who work on the hi-voltage cables wear a protective suit of metal mesh, like chain mail. This seems very counter-intuitive since the metal is a conductor, but it routes any current around the body rather than through it.
If you run over a downed line however, the cable itself may be physically hot and damage your tires. Also, the weight of your car on the cable may improve its electrical contact with the ground causing a lot of rapid resistance heating right under your wheels. Do a good enough job of this and you could get a minor explosion - not car-launching quality, but probably enough to blow your tires.
Cars are indeed one of the safest places to be during a thunderstorm, for the same reasons mentioned above re a cable dropping onto your car. Although in a hurricane, I’d rather be in a cellar.
Shutting down the local power during a hurricane could be done, but there must be good reasons why it isn’t. There’s a lot of board members who will be much better able to answer that one than me - I’ll wait for one of them to come along.