RG post-grad here (it's well-known to some how I lost my academic post but that's another story).
My experience was that the differences are less pronounced at UG than post-grad level, although privately educated students typically have more networking opportunities and connections that will set them up for employment irrespective of their degree quality. I taught many an expensively educated but lacking a spark type who didn't realise that their tutoring likely had more to do with their grades than genuine intelligence. They'd walk out with a middling 2:i into whatever job their parents had set them up with. It's more difficult for others and we shouldn't pretend otherwise, although it's improving.
At post-grad level? Well, it was made clear to me in subtle ways that I wasn't "polished" and not only did I need to be equal to others but better. So I got the distinction, passed without corrections, got the fellowship, the research project and was, when malicious complaints were made, on a fairly typical early career path. The people getting permanent posts were usually, although not always, of a certain background and the stats for working class post-grads and academics in my subject are dismal. It's predominantly white, relatively wealthy, men.
Meritocracy in academia is a myth. Yes, you need the smarts but you're largely banging on closed doors if you're a woman, disabled, working class. Sometimes I feel relieved to be out of a stifling environment. That and the fact universities have become supermarkets with staff being measured on so many things that research quality suffers.