To be fair, that example is only arguably not male privilege.
What would happen if they gave the kids to the male? The male would largely have the higher-paying job, the kids, the money, and she'd be owing him child support. Sounds like making any other choice would be overtly anti-woman.
Male privilege does exist, but it's a natural inclination and nothing we should necessarily feel guilty about. Either males are just better at bluffing, or we really are better at what we do. In either of these two cases, that is a privilege we males enjoy and our male-dominated roles are still male-dominated. That's a privilege, not something that happened by a roll of the dice.
Females have their own privilege, such as making less money than men on the average, but affecting the spending of 80% of it.
In a traditional situation, where a woman stays home with the babies and the men work to support them, imagine if wages were equal. That would mean women who do work make so much more than average that it balances out every woman who doesn't work.