You're comparing a dual-parent household to a single-parent household and claiming that if they made the same amount of money, the children would be equally well off. I'm not sure where you get this idea.
Right. Even when the kids are old enough to be in school full-time, there's still lots of "free" labor that's being left out of the GP's reasoning. A single parent's earning power would need to be significantly higher than a 1950's man's earning power and/or they'd need the same earning power with significantly more non-work hours per week.
Or, you know, if we had any number of institutions in this country which are more common in the rest of the developed world, such as paid childcare and maternity leave, affordable healthcare and rent, real protections against discriminating on the basis of age&gender, etc.