If you look at the actual numbers, it's not the rich having fewer children over time. The people with (some) money (defined here as 2x the poverty rate) have fewer children, period, but they're not causing the DECREASE in birthrate, because they've always had fewer children, and that hasn't changed much. The birthrate among poor people, though, has plummeted since 2008.

It absolutely is an economic issue, and you get what you pay for. If people are not secure in their current status, they're not going to risk poverty to have kids. Children need to be cared for, and that means either professional childcare (which is expensive, since the professional needs to make enough to survive off of a handful of kids) or giving up half of a household's income.