I am far less convinced that life saving interventions are net population creating than I am that family planning decreases it. Written about 10 years ago, but still one of the better pieces on this IMO is David Roodman’s report commissioned by GiveWell.
In places where lifetime births/​woman has been converging to 2 or lower, saving one child’s life should lead parents to avert a birth they would otherwise have. The impact of mortality drops on fertility will be nearly 1:1, so population growth will hardly change. In the increasingly exceptional locales where couples appear not to limit fertility much, such as Niger and Mali, the impact of saving a life on total births will be smaller, and may come about mainly through the biological channel of lactational amenorrhea. Here, mortality-drop-fertility-drop ratios of 1:0.5 and 1:0.33 appear more plausible.
So it looks like saving lives in low income countries decreases fertility, but still increases population size.
From the abstract of David Roodman’s paper on The Impact of Life-Saving Interventions on Fertility (written in 2014):
So it looks like saving lives in low income countries decreases fertility, but still increases population size.