for the very top students in the US, there might be 10 or 20 institutions to choose between

I do think that this is a fallacy, honestly.

Yes, there are choices to be made regarding what environment is most desirable for any student-- PG ones included-- but to assume that a suitable education may only be obtained at the most prestigious of brand-name institutions is wrong, I think.

For some students, an elite school with similar classmates is a good idea. For others, if it means waiting until one is 17-18 and chronologically aligned with an age-cohort, that's not going to work.

Would many parents truly send a 12-16yo student off to live in a dorm at MIT or Harvard?

I think probably not-- not when push comes to shove, they won't. The institutions themselves are not so keen on it either, because they have none of the authority (can't act in loco parentis even if everyone wishes they could) but all of the culpability if something goes WRONG.
Schrödinger's cat walks into a bar. And doesn't.