Universities were always finishing schools for the elite, for like 1,000 years its been that way, and the best ones in the US are here for that since before the country was incorporated, here since almost half a millennia ago!
The last 80 odd years of dealing with the lower class and proletariat at all is a footnote and will be an experiment of folly deep in a university archive for the next 1,000 years as they merely revert to the mean.
Every problem that universities have go away when they go back to their roots. Its the corporate and public sector that tied access to having a degree from these places, that’s not the university’s problem.
And to your point, correct, if the proletariat were only surrounded by themselves they would not want to be there.