I don't think the transactional and combative assumption of gender relations is really applicable, or a constructive framework for this kind of analysis.
Specifically, I dont agree with the premise that the man is "convincing" the woman to have "their" child, and shouldn't "get it for free" . It also seems to imply the father is freeloading if they are the primary caretaker and the mother works.
In my reality, women usually want to have children, even more so than men. In a healthy relationship, it is a jointly owned undertaking, not coercion by one partner. Additionally, the role of childrearing is not seen as a cushy reward and payback for going through the hard work of pregnancy.