I mean contrary to the comment above this is how American universities used to be. They had standing exams... Show up on admission day, take a test, highest scores get admission. Easy

That’s not unbiased though. You still have the question of what’s tested and how it’s graded.

Blind review to same rubric works reasonably well. Same test with same expected learning goals should remove nearly all issues.

Come on. For math/physics/chemistry/CS the exams are easy. Just written questions and answers. Grading is also simple, just check if the answer is correct. For high-school-level questions, all the answers can be trivially checked. We're not talking about PhD-level problems.

It's a bit more complicated for humanities, grading essays is more subjective. But we also have solutions for that: have 2-3 people independently grade each essay, and have a special group review all the cases where the graders disagree.