Surely there must be some sort of test to measure applicants, though?
I don't see changing a test to be a solution. If the problem is not having well equipped parents and network, then the solution has to attack that issue. Which of course is very difficult and spans over decades as it would take generations to change.
However, don't make the test expensive. Basically, make it economically equal to as great a degree as possible. Some will still be disadvantaged of course, but I think at some point that's reality, the fact is everyone is not equal. If everyone is treated equal though, then at least those with value can rise themselves to the top regardless of other circumstance.