The politics of social clubs and small organizations very often are dominated by women. It's just that these groups are often very gendered, and so if you're a male and don't make a conscious effort to cross gender stereotypes in your hobbies, you'd never know they exist.
I remember my time in the Harry Potter fandom, which is ~99% female. Boy, was there politics. I can remember some truly epic [1][2] incidents of Internet drama all driven by people seeking social status through any means of manipulation possible.
But what about the non-gendered "political" things, like trad unions, local political party organisations, universities, other non-gendered clubs, if women are really better at people skills, shouldn't them be dominating those areas in the same way men are dominating STEM? Can you find a group which, at the low levels is about 50:50, and yet women dominate the political/organisational level?
How much counter-examples can you ignore before you say well maybe this "women are good at people skills, men are good at STEM". It seems like people really think "well men are better at basically everything"
Most of those are gendered, but they're gendered in ways that are the "default" for society (i.e. masculine) and so you don't notice them. Traditional unions and local political parties, for example, represent industries and civic institutions that have historically been populated with men only.
In industries that actually are 50/50 or majority female, you do see a number of women in top leadership positions. For example, the president of the national teacher's union is a woman [1], as are the three co-presidents of the national nurse's union [2], as is my alma mater's (which only went co-ed in 1974) college president [3].
I remember my time in the Harry Potter fandom, which is ~99% female. Boy, was there politics. I can remember some truly epic [1][2] incidents of Internet drama all driven by people seeking social status through any means of manipulation possible.
[1] https://fanlore.org/wiki/The_Ms.Scribe_Story:_An_Unauthorize...
[2] https://fanlore.org/wiki/Victoria_Bitter_Wank#Further_Readin...