Sigh, this again. This is a solved issue. Many countries, large countries, with lots of rich people and smart kids have figured this out. Standardized tests for every single student at the same time. Once per year, no exceptions. Best graded students have first pick, nationwide, to all universities in the country. No other factor in place.

Yes, rich kids will on average have the advantage because they are more likely to afford good studying conditions and a tutor, nothing to do about this, rich kids will always have some advantage on average. This system motivates everyone, rich or poor, that what matters most to climb the social ladder (or just not fall) is effective studying.

It really is a great system, you can see what were the grades the previous year for your dream university and course, and give it your all with knowledge that it is solely in your hands, and not depending on alumni, quotas, distance, wealth, etc.