Missing a lot of other pieces from that same source:
In developing countries children are needed as a labour force and to provide care for their parents in old age. In these countries, fertility rates are higher due to the lack of access to contraceptives and generally lower levels of female education. The social structure, religious beliefs, economic prosperity and urbanisation within each country are likely to affect birth rates as well as abortion rates,
Also:
fertility rates of immigrants to the US have been found to decrease sharply in the second generation as a result of improving education and income.
Quite a bit there that contradicts your thesis of people moving to improved economic situations suddenly wanting 6+ kids and the population growing out of control. If people don’t need kids for labor, don’t need kids to support them in their old age, and women are educated and in control of their own bodies, there is reason to think the world might not even reach replacement rates.