And yet through most of history children were considered the property of the father and the wife had no rights where they were concerned at all. The current idea that women should get custody is rooted in the idea that taking care of children is still their primary function while men are going to have other, better things to do.
If most teachers are women, is that because men are discriminated against when applying for positions, or that most men consider the work beneath them -- especially at the primary level? How many female dominated jobs are that way because few men want them -- being more interested in jobs that pay more and carry more prestige? If male victims of domestic violence are afraid to speak up, is it because they are ashamed for women to know they are battered husbands, or ashamed that other men will know and think the less of them? I don't think any of this is a female conspiracy or female prejudice, but about the way men view other men and want to be viewed in turn.
I was watching "The People Against O. J. Simpson" and was again struck by how much sexism and disrespect Marcia Clark had to fight while trying the case. A lot of it was coming from women, too, but men were also making fun of her hair and her clothes in order to ridicule her, although usually men would consider such matters beneath their notice. And when one of her ex-husbands sold a naked picture of her to the tabloids most people thought it was funny rather than an outrage. Why? Because she was perceived as an arrogant strident bitch, simply because in the courtroom she behaved as any prosecutor would, and so (the reasoning went) she deserved any mean thing that was said of her or done to her. A man would never have been criticized for being so aggressive. And that was in the 1990s!
The problem with systematic sexism is that so many people are used to it and take it for granted that they don't even see it, and so think there isn't a problem.