I didn't read the scientific articles but here's my point.
I think the main cause of the smaller birth rates in developed countries are the higher costs of rising a child. Of course, education increases those costs and increases the ability of people to calculate those costs and the consequences of having a son for their own lives and for the child without having enough resources. Less educated people are more likely to have a child they can't educate.
But if having babies becomes free (as in free beer) because of the cornucopia that allow us to stop using money, I don't see another way to control population growth than to plan births, which scares me.