You are generalising and taking too much at face value.
I went to Oxbridge (not going to say which) for my first uni and also interviewed at several colleges, as well as many of my friends also going to Oxford or Cambridge.
1. Neither Oxford or Cambridge university have an open policy to discriminate against privately educated students. This is just a public perception due to political pressure. In fact, in the 1960s, 30% of domestic intake was from private schools, it is now holding at roughly 45%.
2. You are largely correct that you need better exam results if you're from a private school compared to maintained schools. But within that large group, you are actually differentiated mainly by your entrance exam (70% of all admissions), application and interview.
3. Oxbridge has no inate preference for diversity. Academic excellence in general matters far more:
i. Your first critical generalisation is that preferences are expressed from a university level. This is not the case. Every college has its own admissions office and select based on their own college preferences which vary significantly from college to college and can be surprisingly strong and consistent across time.
ii. Therefore, your resume matters far less than you think, depending on which college you apply to and even what subject. For example, for the most competitive subjects, the academic bar is extremely high and in the interview they will be looking for additional indications of the nature of this intelligence such as quick or lateral thinking as well as confidence of expression. For the least competitive subjects, the academic requirement is far more flexible and you can get in largely based on how well you fit in with that specific college's ethos or how much they value diversity.
4. Oxbridge DOES effectively discriminate, not on any illegal basis, but on your "class". Almost all Oxbridge students have a background from upper or upper middle/professional classes, even those from state maintained schools. It is rare to find lower middle or working class kids who go to Oxford. This is likely due to the heavy emphasis on education from an early age as well as the Oxbridge staff, culture and traditions all being from the same cut.
Since this tends to cover a large proportion of immigrant groups, this means you will rarely find such pupils at Oxbridge. For example, there was a notorious article that exactly ONE Black-Carribean student had been admitted to Oxford in 2009. Even the British Prime Minister got involved. Oxford vociferously denied this and said there had been 26 "Black" students out of 3,202.
5. As for Europe, I dare not generalise. This is especially the case since many European countries have lower proportion and diversity of immigrants as well as strongly integrationist rather than the multicultural policies of the UK and US. Also, difficulty of access to tertiary education tends to be much lower in Continental Europe because it is heavily subsidised publically. For example, many courses in many German universities are effectively free, even for foreigners. When things are that different, there's little point trying to compare directly on one specific issue.
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So, Britain does not do "tick box" discrimination even for the most competitive universities. Quite the contrary, at least for Oxbridge, they will take academic quality even if that means sourcing from a monoculture. Only political pressure has stemmed the further reduction in state school access.
In short, you could say that Oxbridge is almost a perfect example of the minimum you could expect without legalised "positive" discrimination in a university system of a multicultural anglo-capitalist society.