“If you don’t take some math contest,”
This may explain some of the differences, the USAMO is not “some math contest”, just like the 100M dash in the Olympics is not a random athletic event. I can say factually that MIT, Cal Tech, Stanford etc. do not look at Olympiads as some math contest.
“Not how a college structures web info or what faculty choose to emphasize”
You have been arguing that applicants should dig deep on what a college emphasizes and values. Isn’t figuring out what some faculty are working on one way to do that, rather than just say hey I’m intellectually curious, admit me! Matching faculty interests with your hs research is a pretty good tack to take in apps.
“Show, not just tell.”
Well isn’t placing at science fair or olympiad showing? I could tell colleges that I’m great at math or science because I have great grades but isn’t winning showing I have more than the typical science applicant? What are some examples of show, outside competitions?