They mean the total fertility rate (TFR), not birth rate. TFR is the better metric for evaluating population trends over many decades.
https://www.macrotrends.net/global-metrics/countries/usa/uni...
https://www.macrotrends.net/global-metrics/countries/usa/uni...
EDIT, also, looking at TFR, it's been bumping around but there was clearly a collapse in the 60s. I wouldn't use that to show that the population of the US is growing healthily enough to support capitalism.