> months of delays for everything

Private childcare is also filled with months (often years) of delays. Expanding on this a bit: if you have a sudden need to get childcare, in much of the country you are not likely going to be able to find something that is convenient and of any quality that is also available within a week or two. If you are willing to spend 2x+ the local median childcare expense, you may have better results.

If you want a quick response, you need either dedicated quick-responders (how are they paid when not responding) or you need a lot of slack in the system (caregivers are allowed 4 kids, say, but most have two or three).

And that also needs to be paid for.