> 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

You provide no basis for the idea that the returns on housing have to drop below the point at which its financially viable to build before housing becomes affordable. You just say "because" then restate your premise.

> Markets don't optimize for "everyone gets some", yet that's precisely what you need for housing.

Just because markets don't optimize for it, doesn't mean it doesn't achieve it.

> You'll always need the government to come in at some point to provide for those left behind by the free market.

Disagree.

Isn’t the initial response to a lack of housing that people consume less housing than they would like, rather than homelessness, eg families with children sharing rooms more than they might like, adults living with roommates, or just people having to live further away from where they would like to be (or moving out of a city altogether)?

I don’t dispute that there are levels of affordability that are bad enough that they start to lead to various forms of homelessness, but it doesn’t seem to me like a fundamental rule that, if some people can’t afford to live alone in a large amount of housing, they also can’t afford to live with roommates sharing a smaller amount of housing, and that the right level of housing prices should also price some people out of those arrangements (ie it demands a pretty high level of inequality if you assume that the market allows typical people to afford to live alone and that sharing can typically reduce per-person rents by half or more)