I'm not sure this is the strongest case you could make for saying this ought to be the responsibility of the government. If people can make more working than childcare costs, then they can make their own decision about how to invest their own time and money. Why does the government need to pay for this, if they argument is ultimately that the wages of women now working is what ends up paying the cost of the service?

I am generally not one to rail against the government running efficient programs for general welfare, I support single payer healthcare in the US and many progressive policies. I do however think we need to consider the line where anything you buy has the government involved in re-distributing resources to make it cheaper for people with less money.

Ultimately, it might be good to allow economic forces to produce an economy again where single income households can afford to raise children, regardless of the gender of the parent who is earning an income. More work doesn't always mean societies or families ended up better off, parents spending time raising kids should not be seen as just sunken cost because we could have gotten tax money out of them.