Everybody has a place to live. Housing isn't scarce. When housing is scarce, people are homeless or living in extremely tiny spaces. They are not. The problem is that people wouldn't be able to afford to own the places where they live.
Housing is only "scarce" in desirable places, and that's simply tautological. The sum of unfulfilled desires is the contrapositive of scarcity.
Housing is a market in which a very tiny percentage of goods are up for sale at any particular time, and the projected prices of those goods are heavily shared among a huge network of people who profit when the prices are high. The owners of those goods are often heavily leveraged, and can't afford to lower the price significantly, and the transaction and mortgage costs push them to raise the price to cover those costs at the next place they buy.
> They can only extract so much wealth from housing before people move for alternatives.
No such alternative, which is why the market is as lucrative to exploit as health care. Finding an "alternative" to health care is just to find another type of health care, and if you find an alternative to being housed, you've just found a new kind of house.