As a counterexample: Dutch universities can’t select for students (for most studies), and annual fees are ~€2k annually (regulated). There will be some bias towards wealthier backgrounds that are more able to invest that money and time, but compared to the US/UK systems I expect it is minimal.
This is not unique, I think systems in the Nordics and Germany similar.
Wealthy students get the best grades since their parents send them to expensive schools designed to maximise this system. They then get into the "best" universities.
What do the Dutch do if there are 10k spots available in a given university and 20k people apply?
Exceptions happen: eg the popularity of CSI lead to a huge boom in people wanting to study forensic sciences. IIRC they were allowed to limit the seats for those studies back in the day.
Similarly, dentistry and doctors have a fixed amount of seats. They have a lottery that is biased based on your high school grades.
Overall I think this system has the better outcomes ‘on average’, but that the true top students are not challenged as much as they would be in a more selective system such as the US/UK
As an unrelated observation, the effect known as Berkson's Paradox would probably trigger along the way which is an interesting and little-discussed factor in this sort of thing. Two factors for getting in to college being wealth/connections and intelligence/ability.
Link for the lazy: https://en.wikipedia.org/wiki/Berkson%27s_paradox