If not, maybe that explanation isn’t a very helpful one.
So it’s not a very good joke, as it implies that there aren’t any other filters, despite their intense concern for reputation.
The joke would only work for schools that famously looked the other way on pretty egregious stuff if you paid them enough -- not on a school that makes a point of heavily filtering beyond the official tuition. (Sorry, "sticker price".)
Side note: It's also kinda funny how the parent expects 45k to be some kind of jaw-dropping, outrageous number, when Americans routinely pay that much for college.
This is not true of Eton (or I guess Harrow or the other public schools).
[0] - Or whatever Boris Johnson got, I can't recall exactly, they were spectacularly subpar but he's masterfully scrubbed them from the internet
https://news.ycombinator.com/item?id=33275396
So… what’s left to make the joke work?
Both of us thought this was hilarious!
Given the life-long opportunities an Eton education affords someone, and the likely backgrounds of that 75%, you’d be reasonable in suspecting “a system of perpetuated inherited privilege”.
When these schools were founded, the alternative was to have private tuition - as opposed to be taught "publicly" and live with children from other families.
(The school I went to was founded in 1513 - but is known as a private school rather than a public one because it was never a boarding school).