"So what about the less prestigious state colleges? Do they get anything? What about Trump University? Who decides?? Is it a private club where the orestigious get to decide who 'counts' as prestigious (thu diluting their own prestige?)"
1. There are plently of state schools with good reputation. UNC, William and Mary, Rutgers, Most of the University of California schools, etc...
2. At a certain level, yes they do, but there will be a cut off (probably somewhere around 200/250). Perhaps schools at the lower end can send 1 representative while schools at the higher end can send 5, at least to the legislative branch, this would probably make about equivalent numbers.
(Btw this mostly restricts the law makers (or possibly interpreters) to be technocrats, those these norms might seep into executive branch practices).
3. Trump University is not even accredited university...it won't get votes.
4. I already told you, aggregate of US News, Forbes, Princeton Review, maybe a few others, and an independent UN adjudicator board, this way you get both US and International perspective considered. Also none of these people are eligible to work for the government (at least the technocratic position) for obvious reasons.
"The non-elite schools would become elite schools. Same problem."
What does this even mean, cause if you have a shift in school prestige but still fundamentally the same distribution, then the original algorithm solves. But if the distribution changes then this seems to imply that education as a whole will become better, I am fine with this.