No, the simplest way is to nuke them. Perhaps you meant “the best way”. There’s nothing simple about increasing the standard of living, at least not the sort of increase required to reduce the birth rate. Giving people more food doesn’t decrease the number of children; it increases it. Yeah, I know, I know, poor people tend to have more children, blah, blah. But giving food now just delays the inevitable. Without food, all those extra children eventually die, and the overpopulation problem is solved. With extra food, the children grow up, have more children, and then those children will either get handouts or die. Eventually the handouts are going to run out, and people are going to die. The sooner, the fewer.
If you decrease family size from 8 children to 6, but there are twice as many people having children, you haven’t helped the situation. Yes, if you can manage to get it down to 2 children, the population will stabilize, but the problem is that along the way you’re going to have to go through 7 children, and 6 children, and 5 children, and 4 children, and 3 children. This isn’t a situation where half of a good thing is half as good. Half of a good thing is bad. Unless there’s a reasonable expectation that we will actually bring these people to Western standards, we shouldn’t keep giving them more food in the hope that they’ll stop having more children.
If their population is increasing, then obviously there is a birth rate between their current one and one that will result in extinction.
Yes, it does. If everyone reduced their birth rate, then everyone would benefit. Sure, each person individually might benefit from more children, but they do so at the cost of everyone else.
If they are substinence farmers, then that means that the adults barely grow enough to survive. Presumably the children are less productive than adults, and therefore they grow less than they need to survive, and are therefore a liability. Where do you see my logic as going wrong?