Opening up to a larger group of students will weaken Harvard's brand. Many of these top institutions are valuable because rich, well-connected people go there, so going there allows one to meet them. There are companies where it is an unspoken rule that they hire only from Harvard, law offices that recruit only from Yale, etc.
I doubt "Harvard-only" companies and "Yale-only" law firms of any considerable size exist. Sure, 3 Harvard friends might start a company, or 4 Yale lawyers may start a firm, but eventually they will have to venture outside their bubble. If you have evidence of an organization with, say, 35+ employees maintaining that kind of exclusivity, please post a link.
