This is a common trope but only kinda-sorta true.
It's definitely true that there is no sign of mass exodus. Net immigration number remain positive (3% ish) but they are well below historic highs, and currently declining YoY. See e.g. https://www.macrotrends.net/countries/USA/united-states/net-...
It's interesting I think that per capita Canada (e.g. another "new world" immigration target) has fairly consistently been much higher, and is currently near 2x. The approx 10x population size dwarfs this in total numbers though (https://www.macrotrends.net/countries/CAN/canada/net-migrati...)