This is almost the opposite of what happens.

The more educated/developed a nation, the lesser their birth rate is going to be.

I understand the "shoulds" but that's not what the data suggests.

In essence, we can't have the pie and at the same time eat it.

The most useful thing education does for children is reduce child-mortality rate.[1]

Sources: https://raphael-godefroy.github.io/pdfs/mali_final.pdf

[1] https://eric.ed.gov/?id=ED503923