You can't escape a monopoly of such size. There isn't a city where housing is cheap. At best it is a temporary situation or your view is biased by working in a above average income field.

Zoning will bever help. Once you make space for more housing, developers will build units to be snatched up for speculation that will be left vacant.

You need the housing market to clear. To release unused units to the market. For that you need land tax or an equivalent.

Or just build huge amounts of homes. It's not impossible. The US doesn't even build as many dwellings as we used to in the 60s and 70s if I remember correctly.

Enable developers to build skyscrapers of 500 square foot apartments until there are enough of THOSE apartments to house literally 100% of the population. Then people can choose to either pay say $1000/mo to live in one of those small apartments, or whatever it costs to buy a single family home. And if people have the option of renting somewhere cheap, even if it isn't ideal, that'll help balance the market. Right now we have situations where people need to pay $2000+/mo for the smallest of apartments.

You can build huge amount of buildings if you are a societ style central planner and build a ton of cheap prefabs outside of the city. Further urban sprawl doesn't help being of commuting citizens and city ecology.

Or if you are a central planner with the power to evict tenants from parcels. Then it's easy, you just lose credibility as a government.

Or under a system with LVT, the land/housing market clears and unused parcels return to the market, solving the problem by itself with just market design techniques. As you can see, LVT is a necessary component for land/housing market to even function without getting stuck in a feudal monopoly.

US home building peaked in absolute terms in the 1970s.

When adjusted for population it's even more down.

https://fred.stlouisfed.org/series/HOUST