>That women are pressed into more caring societal roles, where men are often pushed the opposite way.

except the differences between men and women get bigger the more society is gender equal, societies where men and women work the longest and together tend to be poorer and often worse for women's rights,

this is because men and women are fundamentally different and have different goals in life and in their careers, when women are completely free to go for any job they still have preference.

https://www.scientificamerican.com/article/when-times-are-good-the-gender-gap-grows/

https://www.latimes.com/science/sciencenow/la-sci-sn-women-equality-preferences-20181018-story.html

https://www.bbc.com/worklife/article/20190831-the-paradox-of-working-in-the-worlds-most-equal-countries