Free markets for housing are likely to settle into an "optimum" where some percentage of people cannot afford housing at all, because while construction/rental of housing for them would net a return, it's not worth the opportunity cost. Plus you should not wait for the market to respond to a lack of housing: people will be homeless in the meantime. Supply/demand is mostly reactive, because building for anticipated future demand years down the line is very risky, so most investors don't like that.

Markets don't optimize for "everyone gets some", yet that's precisely what you need for housing. You'll always need the government to come in at some point to provide for those left behind by the free market.