So, what's standing out to me is that you recognize that shortage of supply is a key problem here. Rebuilding slums, renovating old housing stock, and building new units are some of the obvious solutions. What I would like to emphasize is that solving that problem does not in any way require the elimination of markets, and indeed, a big part of why we're in the mess in the first place is due to excessive zoning regulations and perverse incentives caused by allowing people to block construction and building on property that they don't own. A huge reason why governments like China and the USSR were able to manage housing is precisely because they didn't have to listen to Mary and David complaining about how a commie block would ruin the neighborhood character, but that doesn't require socialism. Simply removing local control of zoning would do a massive amount to alleviate this. In addition, because I would also readily admit that housing markets may not adequately provide as well for the poor, I have to add that there is no reason why any government can't build more public housing today to serve those who do get left behind.

The core issue here is that we have not enough housing for too many people. The simple solution is of course to simply build more until we have more than enough, and tell anyone who complains along the way to cordially fuck off. None of that requires the elimination of markets as a concept. I do think you're dramatically glossing over how messy a community-driven process of deciding who lives where would be (have you ever attended a public city council meeting?), but regardless, that isn't even remotely necessary to solve the actual root issue. Indeed, in healthy housing markets in the past (at least in terms of cost), such as 1950s America in the post-war boom, or early 1900s New York City, the common thread is not socialism: it's having more than enough housing for everyone who wants it, such that while the rich of course had their mansions and whatnot, there was still enough housing to go around for everyone.

The fundamental problem here is that, when you have X people seeking half as many housing units, then either the richest half get the housing, or those richest half are going to find some other way to get them by offering the poorer half something else. The only way to solve this is to add more housing units; everything else is just increasingly elaborate band-aids.