I think your questions assume a lot of things:
- Women do most of the shuttling of kids? I've yet to see any evidence of this.
- There are differences between the genders, but 'genetic behavioral differences' is a very broad and deep assumption regarding nature vs nurture and opens a whole other spectrum of basic questions.
- Over representation of a gender in some particular field. Being equal does not mean we all have the same aptitudes and interests.
There’s also some research that over-representation of women as K-12 teachers is responsible for the growing education gap where men are falling behind women. It isn’t just women who are held back by gender stereotypes.