Regulation is not what suppresses production, but the actual profit margins. It's extremely hard to make money building housing because the cost of land and labor are so high. These costs are high because we now live in an advanced economy where land can do much more valuable things than "be housing", and laborers can do much more productive things than "build housing."

Yup. Development companies contract when the housing market contracts. They aren't building houses for the fun of it, they are building them because they believe the 100 houses they build in a hot market will ultimately pay back the land purchase rights. They will never build so many houses as to decrease the cost of a home.

I actually got my home from a developer right after the housing bubble. They confided in me that they were giving away these homes pretty much at cost and that they had to fire a huge portion of their staff because the market was just crap at the time.

Really, the only way to actually achieve that is through the state ownership and build out. The state could also spend a premium on building homes that it sells at a loss or rents at lower rates. But that will be pretty unpopular with the general public.