I don't think the data supports your hypothesis, at least if we're talking recent declines in rich countries from their prior levels and not trying to answer "why does the US have a lower birth rate than Mauritania".

Birth rate declines in the US in the past 15-20 years have been entirely among the poor. Among the higher-income they've been basically stable.

Here's a simple chart: https://www.statista.com/statistics/562541/birth-rate-by-pov...

From 2005-2021, the change in birth rates by group:

Below poverty level - down 25%

Income 100-199% of poverty level - down 15%

Income 200%+ of poverty level - up 2%.