There seems to be a fairly strong correlation between wealth and childlessness, so the obvious guess is that the world is getting wealthier. Exactly why that stops people having kids is a bit unclear to me though. Maybe being a parent is one of those things that is a harder sacrifice to make the more alternative comforts have to be given up.
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.
Economics has a simple answer: public pensions are a classic tragedy of the commons. If the pension system didn't exist, then people would be incentivised to have more children to support them in old age, instead of a free rider problem where everyone relies on everyone else having children.