It is true. That article doesn't really back up the idea that wealthy families have more children and in most of the graphs you can actually see fertility has dropped off with income. Eg, in Figure 1 it looks like the only demographic consistently meeting the replacement rate are low income women in certain cultural groups and a marginal number of wealthy migrants. It is obvious [0] that more wealth -> less children. The effect enormous.
The topmost graph on Wikipedia is about GDP, and when it comes to children-per-village, you’re absolutely right. But down some is the J graph, which looks like the American graph in the link I posted.
[0] https://en.wikipedia.org/wiki/Income_and_fertility