Confounder: if that were the case (society pressures women into making these choices), you would expect that the countries that are most egalitarian and most free would produce the most equal outcomes in terms of chosen profession. Surprisingly (at least to me!), the exact opposite is the case: the more egalitarian the society, the more the genders self-segregate into different professions. The men into more technical ("systematizing") professions, the women into more people-oriented professions.
For example, a recent BBC article asked "Why is Russia so good at encouraging women into tech?"[1] Fortunately, the article contains the answer: economic necessity.
"Most of the girls we talked to from other countries had a slightly playful approach to Stem, whereas in Russia, even the very youngest were extremely focused on the fact that their future employment opportunities were more likely to be rooted in Stem subjects."
So women in Russia follow the money, women in the west follow their passion.
> you would expect that the countries that are most egalitarian and most free would produce the most equal outcomes in terms of chosen profession
I think they will, in time. There are still CEOs around who remember when women were mostly (almost exclusively?) secretaries. I think it's fair to want equality now, but not necessarily fair to expect society to move towards it in such a rapid fashion. Not to mention, even though we're a free society there are still endless reminders that men have these specific jobs, and women have those specific jobs. It's getting better (because people are pushing for it) but films are a usually an example of this: mostly male main characters, most business/finance/tech movies revolve around male characters. Most people to idolize in tech are male. I guess it would be hard to measure, but I'd be surprised to learn this pattern didn't have an impact on a young mind, male or female.
But they don't, regardless of your beliefs. This isn't about expectations or wishes. That data is in, and the most developed and most egalitarian countries (scandinavian) have the greatest self-segregation. And the less developed and the less egalitarian the country, the less the self-segregation, pretty much linearly.
Incidentally, this also explains the drop in female CS enrollment over time. I think we can all agree that the past was less equal than the present. So if your theory were correct, CS enrollment of women would have increased, compared to the past. However, CS enrollment of women has decreased, and people have been twisting themselves into pretzels trying to explain this via the "oppression" narrative. However, no such pretzel-logic is required if you accept the data: men have, on average, different preferences than women, and as societies get more free, those differences express themselves more. Easy peasy.
> mostly male main characters
But why is that? Is it because people don't want to look at women? Or is it because males tend, strongly, towards riskier behavior that makes for more plausibly interesting story lines? I think you are confusing cause and effect.
> Most people to idolize in tech are male.
Most people in tech are male, because of self-segregation, and therefore the ones that people will idolize will also be predominantly male. Factor in the fact that males tend to be much more prone to risk-taking, and the effect should probably be even larger than it is.
For example, a recent BBC article asked "Why is Russia so good at encouraging women into tech?"[1] Fortunately, the article contains the answer: economic necessity.
"Most of the girls we talked to from other countries had a slightly playful approach to Stem, whereas in Russia, even the very youngest were extremely focused on the fact that their future employment opportunities were more likely to be rooted in Stem subjects."
So women in Russia follow the money, women in the west follow their passion.
[1] http://www.bbc.com/news/business-39579321