As a statistician, I’m gonna say we likely have some confounding variables at play here. For one, children of successful military officers/ naval officers probably apply to service academies at higher rates, perhaps to continue a family tradition of military service or because they see first-hand the life benefits of being a long-serving military officer, and successful retired officers (the parents of the applicant) probably are in the upper middle class due to a combo of their military benefits and career options post-military. Second, you may see the phenomena of rich kids/ upper middle class kids wanting to “buck the family” by going into the military for a variety of reasons, especially if their parents did not serve, perhaps to “prove something” or avoid getting stuck working in the “family business.” Lastly, there are highly qualified middle class and lower middle class kids from a variety of backgrounds, sometimes from 1st or 2nd generation immigrant families, who would make outstanding military officers/ service academy applicants, but are perhaps not applying because they don’t think they’re competitive, don’t know how to obtain the Congressional endorsement, and/or it’s otherwise not on their radar as an option. We want more of these applicants but the service academies get so many applicants anyway (and have finite budgets) that proactively scouting and recruiting them to apply is a challenge. So you have some self-selection bias all around.
Conclusion: Not necessarily “rigged for privileged kids.” Probably a combination of self-selection factors and heterogenous process knowledge (i.e. it’s more complex than applying to a regular college so nonpublic insights/gouge give certain applicants a leg up) that result in a different applicant pool for service academies than you’d typically see at other colleges.