So if it is just one, that necessarily will benefit only a tiny percentage of US students. And would this not be need-based? Because high need applicants already have many more than one institution’s worth of opportunities.
I actually think it would very likely be more efficient to have a federal merit scholarship program for almost the full cost of attendance at a private college, say 2000 of them a year. That would be one Ivy-sized-college’s worth of publicly subsidized “excellent” education, available to those without demonstrated need.
Now, how to sell that to the public . . . .