"Rich" countries have tiny birth rates compared to others, you can't ignore that fact by saying "people love making babies everywhere" (I live in Peru btw).
A country gets rich before the birth rate falls, so just driving down the birth rate of a poor country doesn't affect the underlying mechanism. You can't just copy the superficial aspects of a successful society and expect the rest to just fall into place, for the same reason that simply holding elections doesn't magically turn a dictatorship into a stable functioning democracy. You have to address the root causes, if they are amenable to analysis.
So maybe the way to reduce population growth is to make poor areas rich(er)?
Bingo, birth rate is a consequence, not a cause. He's trying to put the cart before the horse.
Money is the greatest contraception. Being middle class typically makes people stress about money, so they worry they can't afford children, but they can freely afford condoms and the pill. The poverty striken can't necessarily afford condoms/pill when they want to have sex.
The middle class is being sexually out performed by the poor and the rich.