The high birth rate is symptomatic of their poverty, culture and corruption.
Why did their high fertility rate 7 child per woman - not lead to an economic boom for them? Why did so much of that population have to live and work in these United States?
It seems using Mexico as an example, that excess population CAN lead to an economic improvement - but only if you have them live and consume resources in another nation and repatriate those dollars back home; and even that was obviously not enough.
Why not?
On the contrary, their high birth rate was the universal norm worldwide prior to the industrial revolution, and some third world countries have taken longer than others for urbanization/modernization to diminish their prior norms to western standards.
Attributing their higher birth rates the way you do here is laughable.