You’re correct. It is an (unproven) assumption of globalists that, by providing maternal and child health to Subsaharan Africa, that the fertility index will fall.
This assumption follows from the economic determinism (Marxism) which virtually all globalists and transnationalists believe in.
And they’d better be right, because the projected population of Subsaharan Africa in 2040 is > FOUR BILLION, half of whom will die of starvation or migrate to Europe and North America.
This phenomenon has been observed multiple times. In Europe, Asia, and the Americas, whereever healthcare is top-notch and children are more likely than not to grow to adulthood, people have reduced the number of children they have. In fact, they have reduced the number too far: the reproduction rate in these areas is below replacement rate. In Africa, where efforts to provide better healthcare still have a long way to go, but are beginning to be effective, the birthrate is now ~4 per woman, where a few decades ago, it was twice that. The replacement rate is about 2.4 children per woman.
Leftists do not want to vaccinate or provide healthcare to Africans because they look at the birthrate and assume that preventing child deaths will result in runaway population growth. I.e., they like the fact that millions of children die every year because they are so fixated on the idea of overpopulation. To them, every dead child is a child who is prevented from contributing to overpopulation (this is also why they love abortion so much.) They do not consider the fact that in every country where good healthcare prevents child death, people reduce the number of children they have--thus the overpopulation fear is probably unfounded.