This sounds more like a problem with real estate than with people. The old Georgist schtick of a ~80% land value tax coupled with a citizen's dividend (i.e., UB

This sounds more like a problem with real estate than with people. The old Georgist schtick of a ~80% land value tax coupled with a citizen's dividend (i.e., UBI for those affected by improvements upon said land) would nip this problem in the bud, eliminating the rent-seeking that causes gentrification. Keeping with your analogy, we might similarly eliminate rent-seeking on prestige by banning legacy admissions and making applications more quantitative (i.e., test-based) and less subjective.