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.

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

I can't find a reference right now but I remember reading in a Julian Stanley/CTY book in the 90's that Harvard had a de facto age limit of 16 after the Norbert Weiner/James Sidis issues earlier in the century? Based on this I believe most ivies have a lower age limit of 16?? Does anyone have better info?

While most Harvard freshman are 17 or 18 years old when they arrive on campus, every year Harvard also admits much younger students. Though these young students prove their academic prowess in the admissions process, their age can pose challenges. Some of these are legal hurdles: none of the students in this story will be of legal drinking age while in college. None were able to apply for internships, vote for their preferred political candidates, or even buy cold medicine at the local CVS during their freshman year. Perhaps more importantly, some young students say they face social obstacles when they arrive on campus.

But for the most part, these students learned how to be the youngest person in the room before they ever got to Harvard.

...

Harvard does not consider age as a factor when admitting students to the incoming freshmen class, administrators say.

“We have no age limits. We’re really looking at individuals on the basis of individual achievement and personal characteristics,” said Marlyn E. McGrath ’70, the Harvard College director of admissions. “Certainly, maturity and self-direction and the capacity to thrive and benefit at Harvard is always a factor, but none of those qualities are associated in any way that we know with chronological age.”

The thing is, by making it not an "official" hard limit, it permits age to be used as a factor in "holistic" admissions instead.

As I said de facto... if MIT has a few 14yo freshmen since ~1950 and Harvard has none under 16 at enrollment that suggests a unwritten rule.

Martin A. Camacho ’14 said he felt apprehensive about fitting in socially; he was not sure if he’d be accepted because he was a couple years younger than everyone else. He entered the fifth grade at age five and matriculated to Harvard at 15.

High school had been difficult, he said, but fortunately his social experience here at Harvard has been a positive one.

“I think people are much more respectful,” Camacho said. “High school was very hard to fit in socially for the first couple of years. Everyone would recognize that I was younger than them.”