In some places (esp those with low access to birth control and sub-par sexual health education), there are too many unplanned children being born to people who do not have the means to comfortably raise a child without being in poverty.
Free/low-cost birth control and better sex ed are proven to reduce these instances substantially.
Often programs like this are subsidized based on income such that if they can afford it, it is subsidized less, or you do not qualify past a certain income. That is one method of managing the program's costs, while still benefitting those who are most heavily affected. This program doesn't appear to do that, but many do.
I am not sure what point you are trying to make though.
I’ll even begin to entertain this argument only once the burden of having and raising children falls on society as a whole, and not individual women. Until then, absolutely not.
That's true that it would be good for more access to birth control, but this is a confusing statement because you benefit economically and socially from each new kid that is born and raised. If child care is socialized, it means that the kids are going to be better taken care of versus the mother is impoverished and (a) doesn't work and stays poor or (b) does work and they aren't taken care of properly.
I guess you can make a malthusian argument that the poors will just replicate indefinitely as resources are made available, but I don't think that's believable at all. You should be focused on making sure those future citizens are properly educated and socialized.