What if the case is that the more choice our women in our society have, the more they choose jobs that deal with people (medicine, politics, management, marketing, social sciences, etc) and the less they choose jobs that deal with things (engineering, computer science, etc)?

Because there's a lot of evidence to suggest that's what happens. As measured by the Global Gender Gap Index , the countries with the highest levels of gender equality, such as Scandanavia, have the lowest levels of female programmers and engineers, whereas countries with much lower scores and more traditional views on genders (such as Iran UAE) have some of the highest levels of female programmers and engineers.

https://www.theatlantic.com/science/archive/2018/02/the-more...

https://www.thejournal.ie/gender-equality-countries-stem-gir...