Not quite, childlessness increases as a society gets materially richer.
There is some correlation between having less children and wealth, but for the 20th century Western World it was because of socialism and the empowerment of women.
Socialism made children too expensive to have.
The empowerment of women supersized their desire for hypergamy, because women still wanted men to be the greater partner but they have less men available to them that are still greater than they are.