Prosperity does cause humans to move towards relatively fewer offspring, but only relatively. If the birth rate exceeds the replacement rate by even a tiny fraction in any demographic, the result is exponential growth. And there are prosperous, well-educated demographics with ultra-high birth rates, such as Mormons.
That assumes the birth rates of other demographics don't fall sufficiently behind the replacement rate to compensate in the short term, and that demographics' birth rates are immutable in the long term. Will the offspring of Mormons maintain their parents' birth rates? Well they're already falling...
