There are very very strong historical precedents indicating that a decrease in childhood mortality leads to a decline in birth rates. You should google this.
"The child survival hypothesis states that if child mortality is reduced, then eventually fertility reduction follows, with the net effect of lower growth of population. In populations living under low socioeconomic conditions, other factors have also been observed. The question arises whether fertility reduction could be influenced by family planning. Bangladesh data have demonstrated that if not a single child died in a family then the average total fertility rate (TFR) was 2.6 children; when 1 child died the number was 4.7 children; 2 child deaths meant 6.2 children; and more than 3 child deaths boosted the TFR to 8.3 children."[0]
I think there are too many extreme exceptions to this correlation for it to be anything but a correlation. For just one among many, look to any of the countless religious groups within developed countries. These range from Muslims who continue have children at a healthy rate, to the Haredi in Israel who are popping out 6 kids per woman on average. Even Africa's blushing there.
The obvious and logical explanation, which fits seemingly all data, is simply culture. An obvious one is extreme consumerism. Extreme consumerism is going to drive big economies which makes healthcare widely available, driving down infant mortality. At the same time it's also going to drive people to spend their lives trying to earn money instead of raising families, driving down fertility.
Without culture playing a largely dominant role it would also have been much harder for countries to artificially manipulate their fertility rates, as Iran and South Korea both did - and later came to severely regret. I'm leaving China out of the examples as they took more of a legal than cultural approach to try to change their fertility rates.
You have examples of cultures with high birth rates but you aren't telling me whether those birthrates increase or decrease with better healthcare/reduced mortality rates.
Technically, that's also an unintended consequence.
Certainly an observation optimism can cling to, but the mechanisms of evolution suggest that eventually most humans will have some form of immunity to this effect (we pass on more than genes)
"The child survival hypothesis states that if child mortality is reduced, then eventually fertility reduction follows, with the net effect of lower growth of population. In populations living under low socioeconomic conditions, other factors have also been observed. The question arises whether fertility reduction could be influenced by family planning. Bangladesh data have demonstrated that if not a single child died in a family then the average total fertility rate (TFR) was 2.6 children; when 1 child died the number was 4.7 children; 2 child deaths meant 6.2 children; and more than 3 child deaths boosted the TFR to 8.3 children."[0]
0 - https://pubmed.ncbi.nlm.nih.gov/1670537/