In SF, zoning limits prevent you from building too high, and you have things like abandoned warehouses sitting unused because they are zoned for "light industrial" instead of residential.
And even IF your awesome new project to increase the housing supply follows all the zoning rules, it is likely to be blocked anyway. This is because the not in my back yard community will retroactively change the zoning laws to prevent the construction of new buildings.
And I'm not talking about governments preventing shitty buildings from being created. I'm talking about developers who want to build extremely high quality luxury apartments, while offering the community 30 percent of the units as rent controlled apartments and still getting blocked, because apparently some people prefer 100 percent rent controlled of a 5 story building instead of 30 percent of a 50 story building.
But I don't think the analogy to cars or other goods is completely valid: You'd also have to mass produce the land to build the houses on, which, from what I know, isn't possible yet today...
Of course you can go another route and try to build many cheap apartments in a small amount of space. That's basically what has been done in Germany (both of them) in the 1950s and onwards: They were building many large, very dense apartment complexes with architecture optimized for easy construction and for providing living space for as many people as possible. (Typically implemented as high-rise "apartment towers")
Cities are still dealing with the after-effects of those projects today: The apartments are typically very small and far away from parks or playgrounds. Maintenance is slso lacking as the owners try to minimize cost. This has made them unattractive for almost everyone who can afford something better. The result of of course that the tenants are for a high part very low-income or "problematic" persons who couldn't get any other apartment. This creates an unsafe environment and makes the apartments even less attractive, leading to a downward spiral...
Modern urban planning had tried to learn from those disasters. We know today that "living quality" isn't just a luxury, it can have a direct effect on vandalism, the crime rate, etc.