> Why does providing assistance have to mean centralized control of what assistance looks like?

I generally agree with you, but often the reason that these programs work economically is that those who don't choose to use them still contribute. There are (at least) three different categories: (1) caregivers who will care for their child themselves regardless of whether or not free care is available elsewhere, (2) caregivers who will find care elsewhere regardless of the cost, and (3) caregivers who will make use of free care if available, or otherwise, care for their child themselves.

I think the group (1) has a tendency to be higher income. It's certainly not true of everyone in that group, but I would wager that a significant number of people in that group do not need the financial assistance. Those people not using the free resource, but still contributing to funding it is what makes it economically viable.