Fun thread!
New theory: The בעל עגלה was very much a male job, everyone can understand that. So “driving” was traditionally something that woman did not do. When cars started being a thing, they were naturally viewed in the same light. Now indisputably driving nowadays is different than the בעל העגלה, but many chassidim hold on tight to traditional ways, and perceive it as a activity unbecoming of woman. Similar to keeping the levush the way it was, a woman shouldn’t drive, she was never the בעל העגלה.