Yet the statistics show that:
1. Those with lower incomes have more kids than those with higher incomes.
2. This trend has existed for more than 20 years, closer to 50 years in fact.
Now that I've pointed this out you seem to want to argue that income is not the only factor that determines fertility rates. Certainly that is true, but given that it's true why did you seem to emphasize it so strongly in your post when it supported the conclusion you wanted to make, but when I point out the opposite only then do you decide to diminish the relationship between income and fertility?
If the relationship between income and fertility is not that significant when I point it out, then it's not that significant when you want to use it for your argument either in which case your entire point kind of falls apart.
Finally as an aside, there is no shortage of countries with either cheap or even free child care, mostly in wealthy countries such as Finland, Sweden, Norway. But guess what? Fertility rates in those countries are incredibly low.
Fertility in Sweden has been higher than the average EU rate for a long time (although I think it's been a bit lower recently), and that has been attributed to the generous family policies and access to cheap (and good) childcare.
480 days of tax-funded parental leave per child (caveat emptor, terms and conditions apply).
Tax-supported child care, max cost per month to parents at about $100 per child.
Parents are allowed by law to select to 75% during the child's first 7 years.
About $100 per month and child in tax-funded child benefit(?).