“There is no question that societies that have reduced child mortality will have fewer children.”

The societies that have reduced child mortality and those that have not are different in other ways, too, such as K-selection vs r-selection. Anytime an r-selected society has received aid from the outside, they’ve become even more r-selected. That’s why western aid to Africa has resulted in an African population boom. If we help them with supposedly good-intentioned health-care, they’re not going to suddenly become K-selected and careful about breeding.