Honestly, I get what you mean. I wrote something relevant a couple of days ago:
The thing is, I also went a private school. Not an egregiously expensive international or 'public' school – but a decent independent school, on a means-tested bursary and scholarship.
I can tell you that the issue isn't where somebody went to school, it's the resources and particular knowledge they had/have.
Despite going to a 'good' school, I always played catch-up when it came to careers. My parents didn't (and wouldn't have known how to) guide me towards a career in commercial law. In contrast, there are students at both private and 'state' (elite grammar) schools whose parents/grandparents exist within a network of people in 'elite' careers. These students can get high-quality work experience at the drop of a dime, and they are taught the exact path to IB/consulting/commercial law while the rest of us are left to figure it out.
Relatedly, this is why I hate 'competency' questions. Asking applicants to "tell us about a time where you overcame a challenge" or "show us how you demonstrated your initiative" is covertly asking which social class we belong to. While overcoming challenges and demonstrating initiative (etc.) is not exclusive to certain classes, the type of answer which firms generally seek will select for a particular type of individual. They're not interested in how underprivileged people overcame long-term challenges (i.e. getting a part-time job to contribute towards household expenses) because these lack immediate-term remedial actions and quantifiable results. They're also not interested in the low-level challenges which you'd overcome in such a part-time role. Instead, they want to hear about how somebody single-handedly improved the efficiency/profitability of a club/society/business (as an intern). And, given the limited amount of executive roles/professional internships, you can guess who's gunning to secure these from the outset!