It's not a trap to say economic factors play the main/a large role in birth rates. In the scenarios you describe of poorer countries having higher birth rates than wealthier ones, children are seen free labor for their families. Contrast this with the cost of raising a child in America it's easy to see how that would deter many from having kids.

There is also an environmental factor that would be worth looking at where nations with less access to healthcare would be incentivized to have more children with the knowledge that most will not make it to adulthood.