IMO the biggest counter-example of the "women are better at working with people / soft skill" theory is politics.
Politics is all about who you know, who has your back, how you can get people to do what you want (the voters, other party members), how you can get people to like you, can you predict when someone's going to stab you in the back? can you predict when to stab someone else in the back? Should you support person X? What will person/group X think if you support person Y/policy Y? It's all about soft skills, it's all about reading people, it's all about working people.
You don't need maths or logic, or numbers,or rationality, or science for politics. I'm sure we all know how bad many polticians are at science/STEM.
If these theory were true, you'd expect women dominate politics in the same way men dominate tech. But that's not what you see at all. Advocates for the "women are better at soft skills, and worse at numbers" need to explain this glaring counter-example. The other theory, of institutionalised sexism against women, fits observable reality better.