Can't speak for all European countries as Europe is very diverse, but in Spain it's perfectly possible (and lots of people do it every year) to get top marks in the "selectivity" (entrance) exam after going to free public elementary and high schools. My partner grew up in a dairy farm, her parents have no degrees, she went to a small school in her small town in the middle of nothing, and she got into a medicine degree, which is one of the degrees with highest grade threshold. And it's not a unique story, it's quite common. So I don't think there is that much systemic bias. Especially if you also factor in the fact that tuition is cheap, or even free if your income is not high, so student debt isn't a thing either.