I feel like there is nothing that makes Princeton stand out vs other HYSPM schools.

Among HYPSM, Yale is the school for liberal arts peeps, MIT is the school for STEM nerds, Harvard is the school for people who want PrEsTiGe, Stanford is the best in the west. What's Princeton's niche? Being ranked #1 on US News? Honestly that's probably the only reason they're still included in the HYPSM acronym.

The lack of graduate schools also hurts Princeton's name brand and lay prestige. A lot of Harvard and Yale's prestige comes from their law schools. A huge percentage of presidents and supreme court justices studied at Harvard or Yale Law, and people hear about this on the news, which improves their prestige among the general populous.