I don't have the sources, but I remember hearing studies that found that when men sue for custody, they are more likely to get it. More women end up with custody, but when men push for it, they get it.

The suggestion that men back off due to fears of being accused of child abuse is unsubstantiated. It sounds like it's just something you've heard from a friend and shouldn't be repeating as gospel. I've heard from a lawyer that men threaten women with loss of custody if the women don't back down on demands for support/assets. I'd like to see actual data on either claim, but I don't think you can put one forth without the other.

The most important thing, though - men and women in America don't even remotely share child care. Just check out the american time-data studies. Women, whether or not they work, spend more time with kids. They do more work caring for kids. They spend less time at wage-earning. So most of the time, they probably should get custody and some financial support.