Could the decrease in happiness also be explained

We can't rule that out.

And if that's the case it would probably be better to focus on equal treatment when within the same roles instead of going back to traditional roles

However, one study suggests that women tend to perceive equal treatment as sexist and unsupportive and regard traditional treatment ("benevolent sexism") as preferable.

Another study also finds that women prefer men's benevolent sexism over nonsexism.

Another study finds that job authority is positively correlated with depression in women but negatively correlated in men:

What's striking is that women with job authority in our study are advantaged in terms of most characteristics that are strong predictors of positive mental health. These women have more education, higher incomes, more prestigious occupations, and higher levels of job satisfaction and autonomy than women without job authority. Yet, they have worse mental health than lower-status women.

The general thrust of the research is that women are most receptive to benevolent sexism (unequal treatment) combined with feeling like they are receiving equal treatment.