As a counter example, I'd like to use the Cuban-American community in the US. All of our families came to this country with little more than the clothes on their backs, starting in the early 60s. Now, half a century later, Cuban-Americans are doing as well for themselves as non-Hispanic white Americans, with equally large proportions in the middle, upper middle, and upper classes. My mom's father had a fourth grade education in Cuba and was a self-made businessman who ran a country store in Cuba and a store-fixtures store in the US. My dad's father was a middle class computer technician in Cuba, who came to the US and was never able to work in his profession again. He worked two jobs at first, sweeping floors and taking whatever other jobs came his way, and eventually rose up to running a hospital department. My father never had his own bedroom as a child. He got almost all of his schooling at very tough (in the high-crime and high-apathy sense) NYC public schools in the 60s/70s. There were no silver spoons in their mouths.
My parents have had successful careers, despite being dead broke in their 20s. They were able to stress the importance of education and hard work to me. I had a great job at Microsoft for two years, which I've now left to pursue a career in law.
And this is more or less the same family story of countless Cuban-American families. So I'm not buying this narrative of a calcified class structure built on generational wealth. The most important thing you can do for your children is not to be rich, but to be good. Values and virtues, not trust funds and special treatment.