[johnnyanmac]

Same way we do now: based on who's parents can donate a new wing for the campus to build.

But sure, it's the same problem with any other prestigious venue. Demand far far far outstrips supply. So they don't really need to pick "the best" students. Merely students "over the bar of quality". There's no problem in the eyes of the venue, so there's nothing to change.

I think the implied assumption in this question is flawed to begin with in that not everyone needs to be at a famous institution to succeed. But if you want my likely bad take: sports coaches actually have a pretty decent method of scouting by... well, scouting. seek out local/state/national talent and nurture them years before an app goes in. If they can build a relationship, that's a personal referral that goes farther than any essay prompt.

It's the most flexible method because scouters can tailor from culture to culture, based on qualities that traditional education metrics wouldn't take into account.

[fragmede]

It's a good idea! The Math Olympiad was a thing when I was in school. Just need to turn that into a more mainstream competition and build up a much larger culture and business around succeeding in it.