Hey, I don't get where the happy medium is either. But these countries need to provide their people with any and all available options to control their birth rate. I would guess the women would take the offer to get on birth control or get sterilized rather than put themselves further into a hopeless situation. 'The sponsorship of a child is just a bandaid for the larger problem. Hopefully it doesn't just encourage them to have more hoping to get more "sponsors" for their families.
The sin issue pales in comparison to the suffering these families go through. You're right on with the society getting on its feet first, so the present population can provide for itself without the continuing addition of more children dragging the efforts to stop the misery down.