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.

[0] https://en.wikipedia.org/wiki/Income_and_fertility