When housing is so expensive, why would low wagies bother if they can't afford a rent. They end up on the street regardless of working.
Housing shortage in just a land shortage in urban areas. There are not that many places to rent compared to the size of the city. Most buildings are out of the market, not suitable for renting out and the owner is cash strapped to get it into a livable condition. Since housing keeps appreciating at such rates lately the owners suck it up with property taxes and depreciation and just keep sitting idle on properties. The houses are out of market because the owner is asset rich, but cash poor, but nobody forces him to sell back to the market.
It's a liquidity problem, not in money, but in houses. When unused housing from asset rich but cash poor owners, who got properties through inheritance, sits idle and never returns to the market, there is no supply and demand dynamic in the market anymore. The market doesn't clear and brings the economy to the halt. (One solution is LVT to force market clearing).