That doesn't have prestige value. Prestige comes from scarcity and the ability to exclude the lower caste.
If people want to play those exclusivity games that's up to them. What's wrong is asking the taxpayer to fund it under the false mask that the entire product is education.
That's not exactly true. Funnily enough, you are extremely dependent on your sociological background. If you come from a poor family and do very well, you'll get a full ride for sure. But if you do well but come from a well-off family that refuse to pay for your education, you are fucked. It's only university attendance that is (mostly) free. you still need to finance housing and life costs. Since most good universities are in expensive cities and student loans are not much of a thing, it is an extremely selective process that targets both class standing (from a money standpoint) and parental implication.
There was a study on one of the most selective school in France and actually diversity of background has gone down in the last 20 years. Europe is highly politicised and it was always about selecting for ideologically compatible behavior. Otherwise education wouldn't need so much government intervention/support, even if said education would be paid for by the taxpayer (everyone could get some amounts of credits, that they could spend on their institution of choice).
America used to do that, but Jewish students started taking (and doing well on) the test, and later Black and Asian students had the audacity to be brilliant too. This led to America's "holistic" college admissions process.
For what it's worth, the USA isn't unique in adapting admissions to reject an unwanted minority. The most interesting mechanism has to be Moscow State University's Jewish Problems: https://arxiv.org/abs/1110.1556
The Chinese national entrance exam (gaokao?) is notoriously grueling, but doing well pretty much guarantees you a spot in a top university. Would have been useful to me, having grabbed a middle-of-the-road SAT score for Ivies but having failed to apply to one. There's definitely a multi-pronged strategy for ensuring exclusivity.
The scarcity in Europe (at least the two countries I'm familiar with) comes from a standardized test. If you don't do well on the test, you don't go to college.