Wow, I looked up the data to refute this, but it was worse than I presumed:

https://www.brookings.edu/research/race-gaps-in-sat-scores-h...

My initial thought was we could just have a floor. We don't need to ensure a perfect score amongst the pool that gets randomly selected, so if there there is minimum that is still "high," and that isn't biased toward outliers from any race, then it would work.

But it looks like the floor we'd need to have a reasonable representation would be something like a 500/800 math score. That is much too low, at least for elite schools. Choosing something like a 700 cut-off ends up being very racially selective.

Maybe that's what it should be, then? I wouldn't expect to get in, displacing a student with a higher score, just because I'm white.

Making more universities like Caltech sounds wonderful to me.

I think you misunderstood. A Random acceptance above an achievable standard (e.g average marks of 80%) would reduce racial disparities. It doesn't matter what races are the top performers, it just matters how many people can achieve the base standard.