Sure, and lets make the speed limit 5mph and avoid 1.3 million vehicle deaths every year. It'll take people longer to get places but we know what's best for them
People will try to game the system any number of ways, from faking scores to faking backgrounds. You don't say your criteria are broken b/c people are cheating - that's affirming the consequent
If your goal is equal opportunity you can look at socioeconomic class without considering race. Fundamentally the idea that race must be considered to achieve diversity is an inherently racist idea - you don't believe some races will get in without help.