There is maybe one country in the entire world where births are going up. The gist of your argument is correct, but it’s not “births going up” but “the birth rate is still higher than in rich countries, even though it’s been going down for a while”. The general pattern that all countries go through is from high birth rate and high death rate to low birth rate and low death rate. It’s just that death rate goes down way faster, because it’s mostly dependent on technology, and birth rate going down requires significant behavioural changes. The population explosion in poorer countries is because death rates have gone down way faster (lots of change arrivibg suddenly) than they did in the rich countries (where thus all happened long ago, as the technology was invented). Look up population transition if you want to know more.