I edited the comment — “celebrity” might be the wrong word, but I was thinking that they might care more about having a noble’s kid attend than the 45k, and, like you just implied, they’d reject nouveau-riches even if they had the money, especially if the riches came from a source that’s frowned upon.
So it’s not a very good joke, as it implies that there aren’t any other filters, despite their intense concern for reputation.
They need to have enough truth/insight to be funny. When your premise is that Eton only looks at cash-in-hand for admissions -- pretty much the opposite of the truth -- that doesn't qualify.
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.
So it’s not a very good joke, as it implies that there aren’t any other filters, despite their intense concern for reputation.