I do think men are better than women at some things on average (chess, programming) and that women are better at some things than men (verbal ability, including foreign languages).
Women and men are different; most women have a maternal drive to have a child/ren that probably consciously (and subconsciously) affects them in ways which do not remotely register with men. They're just wired differently.
I'd say you've both captured the conscious and subconscious messages our society tells young girls, which is why they don't go into programming. "Oh, no, I'm a girl, I'm wired differently, I'm better at verbal skills than programming..." etc. None of these tropes have anything backing them stronger than anecdata and social selection bias, but they're out there nevertheless.
Granted, it's not "a woman's place is in the home," so it clearly shows how we've made progress on equality, but it also shows there's a long way yet to go.
