Yes, women are in average higher in trait agreeableness[1][2]. This is well studied cross-culturally. There's still a huge overlap between men and women in most traits so it's not useful as a rule of thumb of how people as individuals.
It can explain average population outcomes like men on average haggling for better wages as you said.
It can also explain outlier outcome, for example the most disagreeable people are mostly men. This makes men more likely to end up in prison. Which doesn't mean that the average men is much more likely to end up in prison than the average woman. But the type of person who is more likely to end up in prison happen to be mostly men.
It can explain average population outcomes like men on average haggling for better wages as you said.
It can also explain outlier outcome, for example the most disagreeable people are mostly men. This makes men more likely to end up in prison. Which doesn't mean that the average men is much more likely to end up in prison than the average woman. But the type of person who is more likely to end up in prison happen to be mostly men.
[1] https://en.wikipedia.org/wiki/Agreeableness [2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3149680/