It’s far from “quick,” because as you recognize licenses and degrees don’t translate. I know a few people who worked their way back up by getting degrees in the U.S., but that takes many years. My cousin’s mom redid med school and residency, during which time my cousin grew up in a two bedroom apartment in Queens with her parents and brother. Note that the study I linked above compares the income of the household in which a child grew up to that person’s income in adulthood. So a kid whose parents “quickly found a white collar job” wouldn’t be categorized as having grown up in the bottom 20%.
Also, your anecdotes are strong evidence of meritocracy! If a doctor or lawyer in old country can start out as a bellhop in the west and then work their way back up to being a doctor (like my uncle) that’s meritocracy!
My mom is from a rich family back in old country, but extensive third-world landholdings don’t count for anything in America. And her master’s degree in chemistry was worth nothing in the U.S., especially given her limited English skills. So she worked in retail her whole career. At her earnings peak 15 years ago she was making $120k+ as a floor associate at a furniture store.