I find it very hard to understand how driving a wagon and car can be equated. There are several fundamental differences. Wagon drivers sit outside the vehicle, open to the entire street. They get spattered with mud and clothing gets blown with the wind when driving at high speeds. It was also usually part of the job to take care of the horse as well.
Car drivers sit inside along with the passengers, are protected from wind to the same degree, and don’t generally do dirty work. Early cars needed to be started with a crank on the outside, so that was a tznius concern. Women starting a car was something very noticeable. Cars of today are vastly different.