All land is owned subject to the laws that govern its potential uses. Land that is open to more uses is more valuable than land that is not. Land that can be subdivided is worth more than land that cannot. Allowing people to build more houses on "their own land" is a wonderful gift to land owners. It increases the value of their property. But that exacerbates the problem. A trend towards greater freedom to develop real estate means it is, yet again, only a more attractive investment vehicle.
Rather than "let them build", perhaps we should consider "thou shalt build" or "thou shalt rent". Laws that demand certain real estate owners do X or Y could not only increase the number of rental units but also diminish the attractiveness of unoccupied or unused real estate as an investment vehicle.
what kind of internet is available in the area? How close are you to grocery stores? Is there a HOA that you have to pay dues to? What's the weather like? Is the home up to date or is the wiring and plumbing likely to need a lot of work? How big is the lot? How walkable is the neighbourhood? Is it near a smelly dump? What's the crime rate like? What are the schools in the area like, how close are they? How much access is there to transit?
It's not as simple as just "Build more housing" society also has to adequately service the people who want to live in that housing. If our lower income housing is all just bare minimum roof over your head and no access to anything then no one will want to live there, which also means that supply will not affect the demand on housing that has better access to amenities.
We can also be quite certain that providing the bare minimum housing pulls in demand from those more luxurious options (wants) to the more economically practical option (needs), because the bare minimum option was never available. We'd be giving people more choices and at least some would take it.
Then if you have so many people that "demand" a certain standard of luxury housing, then they will have to compete and pay the premiums for the actual limited resources like lakefront property or great schools (living near other like-minded wealthier people). That's the realities of scarce resources, and is fine, since there's not much we can improve there. But the major gap is providing the bare minimum of housing which we are currently not doing.
You'd have to increase density, and most places it will be difficult to displace people from their homes to tear them down and redevelop them.
Only to the extent that existing regulation favors the advantaged more than the market naturally does, which tends to make that approach quite limited in potential.
Well, we could start with the fact that there are roughly 25 empty housing units in the U.S. for every homeless person.
Housing a very basic human need.
There are zoning codes everywhere, environmental review, city permits, etc, that the small amount of successful projects will of course be the minimal low hanging fruit. Literally all of this problem is "the state."
Let people build buildings on their own land. It will supply far more housing than any state backed affordable housing and will break the housing-as-investment pitfall by breaking the supply-constraint that causes people to treat it that way.