Child support is a funny thing. I'm working for my dad's law firm right now -- we're one of the bigger players in the family law game here in town, so I see a lot of child support cases (one of our cases was in the national media just recently). In my time here, I've observed that men and women at think of child support very, very differently.
Women tend to see it as a right, something that they're owed, something that the man pays -- I've even heard a woman who just lost custody on a motion to modify custody exclaim "Child Support?! I thought that was a man thing!" -- no shit.
I've never seen a woman offer to forebear on collecting child support in exchange for custody -- not ever -- or at least not without the man terminating his parental rights (in Oklahoma, you can't agree to not collect child support, but many try to make deals anyhow).
Men see it as an obligation to pay the woman, and 9 times out of 10 they will GLADLY offer to refuse child support in exchange for custody.
Those are my observations based on a pretty significant caseload in central Oklahoma