Is it avoidable having a hereditary elite? It seems like every society develops one. It even happened in the supposedly classless USSR and they had a name for it the nomenklatura https://en.wikipedia.org/wiki/Nomenklatura. Even in the US there has existed groups with hereditary and elite status but there has been a lot of social change over the past 100 years. The American problem is deciding who that elite should be now. I think a lot of the conflict and turmoil we are having is caused by different groups fighting to establish themselves as the hereditary elite.
Peter Turchin's theory about elite overproduction might have merit applied in our context. We have too many people vying for too few elite seats and the losers seek vengeance. Admissions into elite schools appears to have become a battleground.
The essential reason America exists was we said "hell no" to a parasitic, hereditary elite, the British "nobility". So I think it's very in keeping with the American experiment to prevent the reemergence of that sort of elite. I'm not sure we should have an elite at all, but to the extent that we do, I think college admissions should absolutely not favor people based on wealth or family ties.