They do slightly, according to this study at Northwestern.
The meta-analysis showed that women are more likely than men to use leadership styles that other studies have shown produce better worker performance and effectiveness in today's world.
Specifically, women were more likely to be transformational leaders, defined as those who serve as role models, mentor and empower workers and encourage innovation even when the organization they lead is generally successful.
My experience has been that female managers tend to reward for performance and give praise more than male managers. That definitely builds employee confidence and loyalty, which in turn increases productivity. The study ends with this surprising bit:
And the glass ceiling itself may produce more highly skilled female leaders. Research shows that higher standards are often imposed on women to attain leadership roles and to retain them. Because transformational leadership constitutes skillful leadership, women may be more skillful leaders than men because they have to meet a higher standard.
So perhaps the difference found in this study boils down to the fact that because of the glass ceiling women only make it into these spots if they are superb leaders, while some mediocre men make it via the good ole boys network.
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I think there is very much truth in what you say about women having to have been better than the men that they competed against to ‘get on’; these are after all not average women. These women do a job because that is the job that they want to do, not because it is simply a vehicle for their own status progression.
However, some women who ‘get on’ do seem to feel that they owe it to other women to help them get on too, - in a way that most men (particularly alpha males) never do unless there is something in it for them. They will thus act as mentors to other, often younger, women of inferior rank (yes I’m a male!) and meet them socially and individually in a way that most men would find unacceptable. Unfortunately the consequence can sometimes be that the younger ’second fiddle’ is promoted into a job role that is beyond their leadership abilities. The public sector in the UK has many such examples of such affirmative action inspired promotions.