RAC cover the person not the car.
So do the AA.
Actually they give you the choice, person / couple / family is the norm but you can have car cover if you want. There are restrictions on car cover, one being that it must be less than 10 years old which rules it out for me.
I've been with the AA for 8 years. I've called them out a few times and they have always responded very quickly. The last occasion, earlier this year on a back road beneath the South Downs, they got me running again in less than an hour after calling them - so I can't fault their service.
My only gripe is the hike in premium they make every year, which means calling them up and haggling.
~iw