> some people who would rather take care of their own children are pushed to take a job and have someone else watch their child

You could say the same about the taxes used to pay for public schools.

> fail to see why that's superior to a universal childcare subsidy

For one, it doesn't free up labor. Two, most systems I'm aware of let one take a childcare subsidy and use it at home. Paying providers by default makes administration easier by reducing the number of payees. (There is also an oversight cost with private payments.)