I don’t see why not. America has a long track record of up from nothing elites.
> Lloyd Craig Blankfein was born in The Bronx borough of New York City to a low-income, Jewish family on September 20, 1954. His father, Seymour Blankfein, was a clerk with the U.S. Postal Service branch in Manhattan and his mother was a receptionist. He was raised in the Linden Houses, a housing project in the East New York section of Brooklyn.
> Sotomayor was born in the Bronx, New York City, to Puerto Rican-born parents. Her father died when she was nine, and she was subsequently raised by her mother, who worked long hours as a nurse to support the family.
> Andrew Carnegie was born to Margaret Morrison Carnegie and William Carnegie in Dunfermline, Scotland, in a typical weaver's cottage with only one main room, consisting of half the ground floor, which was shared with the neighboring weaver's family. The main room served as a living room, dining room and bedroom.
> Lincoln was born into poverty in a log cabin in Kentucky and was raised on the frontier, primarily in Indiana.
> Feynman was born on May 11, 1918, in Queens, New York City, to Lucille (née Phillips; 1895–1981), a homemaker, and Melville Arthur Feynman (1890–1946), a sales manager.
> [Tom] Cruise was born on July 3, 1962, in Syracuse, New York, to electrical engineer Thomas Cruise Mapother III (1934–1984) and special education teacher Mary Lee (née Pfeiffer; 1936–2017). His parents were both from Louisville, Kentucky, and had English, German, and Irish ancestry. Cruise grew up in near poverty and had a Catholic upbringing.
On the other side of the coin there are plenty of second and third generation people running medium or occasionally large companies, Hollywood famously has a nepobaby problem, and there are other bastions of inherited privilege. But we are by no means dominated by certain families the way some other countries are. We have a chance to become even less dominated by breaking the remaining pipelines, including legacy admissions (inheritable dual class stock is another one to look at.)
I think a super rich kid who can't quite get into say, Harvard, but instead goes to some other school, is not going to be unemployed or pushing paper in middle management, they're still going to work for the family firm, start a business with family money, or cross-pollinate among the other elite families.
Furthermore, if the would-be legacies can't get into Harvard and Yale, the most likely outcome I foresee is that they start to cluster at other schools (say Amherst, Tufts, BU[1]), gradually shifting the character and reputations of those schools and getting us right back where we came from.
I don't really think that is a bad thing, and think it's probably best to stop doing legacy admissions. But I think there's no way this will result in reshaping of class in our society to where the elites are usurped by a bunch of smart, diverse (merit-admitted) kids from the wrong side of the tracks. Best case it gives a boost to the best non-Ivy schools at attracting the descendants of the Harvard and Yale set, potentially to the point of altering society's definition of which schools are the most elite.
[1] forgive any errors in my choice of schools – I just googled top universities in New England and skipped over ones I know are Ivies.