What it does do is that if the custom (it used to be in Israel at one point) that only a man drives then it becomes somewhat of a לא תלבש issue. The same way you understand that a woman should not be driving an 18 Wheeler, although there is no specific issur..
This is correct. But there are additional halachic issues as well. There are sh”ut written about three psakim prohibiting women from driving. There are multiple issues. They were cited in the previous women driving threads here.