My take on the topic is, give where your heart tells you to give. Do the best you can. Are you going to save every child, extending the human population by a million or so person-years per year, before you save a single animal? Cause I'm not.
As far as which kids, my reflex is usually to donate to the Red Cross, because they've been doing the charity thing long enough to know how to cope with corrupt governments and what not. I'm sure everyone has totally altruistic motives, but I hate the thought of charitable contributions rotting in a warehouse or being sold to fill some tinpot's pocket. A certain amount of that is inevitable, but it should be limited as much as possible.
While I can follow your logic, I think that in practice these "save a kid" ongoing donations usually ensure that that child receives at least a decent primary school education. And it is demonstrably true that females who can read and write have FAR FEWER less children than illiterate females. In fact, get a nations children to a certain literacy level, and the birthrate drops to at or below replacement level without ANY government intervention in reproduction.
So I think that "save a kid" would probably have a good chance of leading in the long run to fewer offspring in that child's community overall.
I favour Community Aid Abroad because they guarantee that no more than 30% of their donations go into administration, and they specialise in long term on the ground comunity projects, that are far less likely to be stolen by armed tyrants. Things like wells, small damns, schools, farming tuition, that sort of thing. They do do more stealable items like goats and mosquito nets, but meh.