> Because having each parent have to go to a privatized daycare utimately costs more, for a worse service than it being a public service.

Why would it work in such a way for this when it works the opposite for everything else?

> It also reinforces social segregation, where wealthy parents can afford to put their children into the fancy daycare, while poorer ones either don't (which is horrible for the children), or have them in run down, overcrowded day cares.

If the subsidy is enough to pay for the "fancy" daycare then everyone can afford it because everyone gets the subsidy. If it isn't, you haven't done that to begin with, because then the public daycare is underfunded and the wealthy parents use their money to opt out of it regardless of whether it's operated by the government.