As far as I know, the women and girls are NOT told that driving is a Tznius issue, just that it was never “oongenimen” (accepted) or “eingefirt” (instituted/introduced) by them. It seems to be a minhag that they uphold because “my mother din’t drive and her mother didn’t either….”
I was told flat out that the reason is it not considered tzniusdik for women is that driving motions involving the moving of the legs from pedal to brake, can potentially incite improper thoughts in the women. It may be so, but try telling that to the mothers driving screaming kids in a carpool.