Bill Gates' mom was on the board of directors of a bank and gave him an intro to IBM execs. [0]
Bezos parents invested 250k to fund Amazon. [1]
Upper middle class is generational wealth already if it passes a generation.
[0] https://www.cnbc.com/2020/08/05/how-bill-gates-mother-influe...
[1] https://www.cnbc.com/2018/08/02/how-jeff-bezos-got-his-paren...
And none of them were "heirs who own the majority of stock" nor support your claim that "just because the companies change doesn't mean the owners do".
Edit to reply: You don't need to tell me about the social mobility of people living in poverty. I was born into a family of construction workers, and escaped poverty by learning to program. I know from experience that new businesses create opportunities to escape poverty, even if that isn't measured or sufficiently weighted by the index you cited.
I'm not as concerned as you with the various strata of richness and the mobility between them.
I'm much more concerned with the social mobility of people living in poverty, which is pretty bad in the US compared to Europe. [0]
Key factor in that? Unions.
[0] https://en.m.wikipedia.org/wiki/Global_Social_Mobility_Index
Zuckerberg bought Instagram from Kevin Systrom who went to Middlesex high school, which costs at least $55,000 a year.
At this point it becomes a parlor game of who is rich, and who is "upper middle class", which apparently means spending $55,000+ a year on your kid's high school.
Edit: Zuckerberg's parents were a psychiatrist and a dentist. Well paid jobs but still jobs. I'd hardly call it generational wealth.
Do you think the child of a poor single mother trying to make ends meet is given the same opportunities?
Steve Jobs father never graduated high school and fixed cars for a living.
Larry Ellison grew up in an apartment in the South Side of Chicago.
Starbucks founder Howard Schultz grew up in a public housing project.