> Renting a private room was possible on nearly any wage 50 years ago, and the only reason it seems out of reach for many now is because purchasing power has been slowly stagnating for decades

50 years ago, in high cost of living areas, you could rent an SRO, but now they're either banned or practically banned because they're strongly disincentivized against. Combine this with not building enough new housing and you get a recipe for rent increases. Even if a minimum wage works as intended, it can only subsidize demand, which would do nothing when the bottleneck is the supply.