What's interesting about what you said is that if one were to look at the CEOs of the Fortune 500, this idea of classism breaks down.
CEO of IBM: Ginni Rometty -- grew up in a single parent home, with a mom that worked multiple jobs. Ginni got in to Northwestern and studied computer science and electrical engineering. She certainly didn't have any upper-class advantages. She went to public school.
Geisha Williams, CEO of PG&E (part of the Fortune 500) -- daughter of Cuban refugees -- literally arrived in the US with nothing, her father was a political prisoner in Cuba before being released. He worked multiple jobs to support his family. Geisha started her career as a residential power auditor and worked her way up to CEO. Hardly a "silver spoon" story.
Rex Tillerson (former) CEO of Exxon -- one of the largest companies in the world. Picked cotton and worked as a janitor before getting accepted to the University of Texas with a band scholarship. He joined Exxon as a production engineer and then worked his way up to CEO over the next 30 years. Hardly an "elite."
Tim Cook -- son of a dockworker and a pharmacy worker. Went to Auburn University (hardly an Ivy League, "elite" school.) He worked at IBM from 12 years working his way up to the director of North American fulfillment while earning a Duke MBA at essentially night school.
I could tell these stories all day. There certainly are some elites who do go on to be CEOs, but there are even more CEOs and self-made millionaires that aren't Boston Brahmins or cousins of the Kennedy family.
This whole "classism" idea is promoted by three types of people:
a. those who are actually ignorant about American business and its leaders. Spend some time actually learning the backgrounds of American business leaders and you'll see more "State U" than "Harvard U."
b. those from other countries where a public school girl from a broken home would never become CEO of IBM or a redneck janitor on a marching band scholarship could never become CEO of Exxon -- these are often Europeans where, in places such as France -- if you aren't a male who went to a Grande Ecole, you will never be CEO of a major French company. CEO of Groupe PSA (France's largest car maker) -- Carlos Tavares, graduate of Ecole Centrale Paris. CEO of Renault? Also a Grande Ecole graduate. Former CEO of Airbus? Louis Gallois, also from a Grande Ecole. CEO of LVMH? Bernard Arnault, another Grande Ecole graduate. The "egalitarian" French have nothing but elites leading their top companies. In the US, the most valuable company's CEO? Tim Cook -- Auburn University -- a state school. Rex Tillerson? University of Texas. CEO of Walmart (the top company on the US Fortune 500 by revenue) -- University of Arkansas and University of Tulsa. CEO of United Health? Dave Wichmannm, Illinois State University with an Accounting degree -- not even an MBA and he's the CEO of the number 6 company in the United States. CEO of United Airlines? Oscar Munoz, the oldest of 9 kids in a Mexican-American family in California and the first to graduate college in his family. Some elite. What "classism" helped Munoz? He's the CEO of the world's largest airline.
c. the type of person that is angry at their own life, their own bad decisions or their own lack of success that they want to create a political narrative that supports the "evil capitalism" trope. This is the type of person that feels like they have been wronged by life and society and rather than looking at their own failings, instead attempt to blame some boogeyman. Maybe their daddy didn't love them enough. I don't know. But these are negative people that claim conspiracy when there is none. They themselves just don't think they're good enough and they need an excuse to justify it.
Now, if we were talking about politics -- THEN I might agree that classism has some influence. When we have a brood of Bushes, a variety of Clintons and an infestation of Kennedys, then it's pretty clear classism does have a real impact. But in the C-Suite -- I would argue that the evidence suggests that classism isn't a factor at all, instead it's about hard work, tenacity, luck and intelligence.