Came here to post that documentary (the entire series of which is freely available, btw[1]). I think you misinterpreted it slightly, however. The documentary presents evidence (strongly denied by the Norwegian social scientists interviewed) that male and female brains are wired differently, to be good at and enjoy different activities. It even goes so far as to show studies that link receiving an excessive amount of the opposite sex's hormones during prenatal development to developing the stereotypical interests of the opposite sex early in childhood.

When a very egalitarian, gender neutral society like Norway exhibits more "traditional" gender distribution in the workforce than a less egalitarian one, the Norwegian social scientists thought this must be because of institutional sexism, or cultural brainwashing reinforcing gender roles, or whatever other excuse you've heard a million times. The documentary instead posits the idea that in a society where you are truly free to do what makes you happy, maybe it's normal that less women than men would want to be engineers and less men than women would want to be teachers or nurses.

Whereas in "somewhat free" societies that have higher numbers of women in "male" jobs, these societies are usually much less affluent than, say, Norway. So of course people will end up doing jobs that they may not particularly enjoy, they have to to survive. It's not mild social pressure "daring" women to go against the grain or anything that could be imported, it's economic pressure forcing people to take whatever jobs are available (or indeed, having jobs assigned to them by a planning committee).

Humans are adaptable, of course. When an intelligent person is forced (directly or indirectly) into a role that they may not have chosen themself, they will most likely learn to be competent. And both intellectual outliers and people (as mentioned above) with prenatal hormone mismatch can account for people that do exceptionally well in a role dominated by the other gender. But the question is, are you really making society better if you are forcing average people into certain roles to fulfill your political goals? Or are you increasing the amount of unhappiness in the world and perhaps even driving down the average competency in that field?

[1] https://en.wikipedia.org/wiki/Hjernevask

If this holds up to scrutiny maybe a better way forward would be to try to increase the status and pay jobs that typically attract women?

Note: I am not saying girls shouldn't code, quite on the contrary I'd hope that all kids, regardless of gender, get more practical subjects earlier in school: coding, woodworking, low-voltage electronics, mechanics, cooking & housework etc etc.

I have to say that while the Scandinavian countries are seen as bastions of gender equality, my personal experience has shown some extraordinary sexism there in academia, far far far more than I've ever experienced in mathematics or physics in the United States. Like men straight-out telling women they don't need to get that science PhD because they'll just drop out to have babies anyway, since all women just want to be mothers above anything else, or a friend being explicitly denied a grad fellowship because her slacker male colleague "has a family to support" so deserves it more (despite fewer publications, fewer grants, behind in the program, stopped showing up to the lab for a while). Scandinavians have an inflated view of their own gender equity (and so does everyone else).

Humans are adaptable.... look at all those women who were competent teachers or nurses even though they'd far prefer something else, forced by economic pressure to take the only jobs allowed...

This "nurses or teachers" stereotype is awful. There's plenty of women who are both smart and not people's person.

Pretending that there is this one axis of "free vs. non-free" and that Norway represents best how people behave free of constraint is ridiculous. You are doing the mental gymnastics you accuse your opponents of, in service a thinly veiled promotion of the "men work, women care" theory.

Was there extra social pressure to stay out of STEM in in the less free societies?

A couple plausible alternative explanations that quickly come to mind are greater reliance on aptitude testing and having more motivation to get a job that granted more freedoms.

(I don't have any idea if those are true, it just seems that there could be lots of things that would explain the difference)