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)