My firsthand experience says this is wrong. Latin American countries experienced massive population growth in the last hundred years and most people aren’t farmers. There isn’t enough arable land in most Latin countries that isn’t already occupied. Infant morality rates have fallen pretty predictably as well.
People like to fuck. They hate wearing condoms. They probably always will. Education (of women) seems to be the key to lowering birthrates and is probably the single most important aspect of our near term future. Forget the US, birthrates here are falling and have been for most of my life.