I firmly believe that birth control is the key to reducing childhood poverty. Can we ever eliminate it in its entirety? I very much doubt it.I'm not qualified to tell anyone they should or shouldn't have children but if you have them you have to meet the consequences and so do your children.
So a great way to remain poor would be to have children within a single-parent household, in which the parent cannot take up significant employment. Or to bring children into a conventional two-parent household in which neither is employed.
We must all accept that poverty's not a crime, as it would be quite evil to criminalise the poor. But therefore nor can it be a crime that a person makes choices that leave them poor. Nor that they have children who will also be poor. But it isn't my responsibility to make them not poor any more. We're all free to make choices, and some choices will be bad.
I agree with you on not letting the government get involved. Any government capable of interfering with our reproductive choices is capable of doing much worse to us.
Education and raising awareness among the poor is how I believe the situation should be handled. In poor nations around the world there are millions of child bearing aged women who desperately want to learn about family planning. They also need access to affordable or free birth control. In their undeveloped societies they know their country is too poor to help them so they are on their own unless caring people from developed nations can help.