Important clarification : to the best of my knowledge, the Rebbe’s response regarding women driving only applied to women who had moved to a community where women didn’t drive, not merely visiting.
I remember hearing this together with another story that one woman drove anyway I guess because she figured it can’t be that bad if our Rabbanim pasken that its fine. End of story : unfortunately she was involved in a bad car accident r”l.
The Rabbanim who don’t allow women driving base it on the fact that a woman isn’t meant to be “out there” in the world like a man. I don’t know more as it isn’t our shita in lubavitch but that’s what I recall hearing.