I can only speak about the humanities, not the sciences. So, to start with: "better research" has nothing to do with it. That's not the point of elite institutions.
Rather, they are about what the Germans call Deutungshoheit -having the sovereignty over the definition of thought. If you or I have an idea, no ones give a mouse's crap, while if you are associated with a prestigious university and say it you will go down in history for having said it.
That's why Harvard & Co are focused on the sensational and the general. They want to get credit for the big topics and issues in life. And they have unique and easy access to top journals, publishers, and the media. In fact, they are always thought after (while the average academic has to try very hard, too hard). Important governmental and non-governmental institutions, even religious groups, establish offices in Cambridge, for example, such to have proximity to Harvard and MIT brands.
Just presenting a talk there is more valuable than tens of thousands of dollars in marketing. Those academics' opinions matter a million times more (than the rest) precisely because they are Harvard, Princeton, MIT etc. professors -and NOT because they are so original or unique. I love it how the research above concentrates on citations etc. That's laughable and besides the point of education, which is not about knowledge but about privilege.
A professor at some no-name college can write whatever he wants, he will never have access to the elite and powerful in academia. Last, if we look at the human race, the perks, privilege, and opportunities to those few chosen ones with a degree from (or a position at) those elite institutions are (almost) limitless.
Everyone wants to be associated with their brand. They feel like gods and goddesses, and in a way they are. So, Why climb the greasy pole? Because atop there lies Valhalla.