Doesn’t that still track? Having children makes it harder to get/stay rich, not having children makes it easier.
There's also the whole "not being aware contraception exists because you couldn't go to school" thing.
Sex is free, fun, and easy. The less education you have the less likely you are to understand effective birth control. The less wealthy you are the less you have to spend on contraception. Those things together with the easiest and cheapest form of entertainment would seem to cirrelate with a higher birthrate in lower income brackets.
I cant find any studies that support what Im saying right now, but the logic holds.