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