You do indeed read it almost exactly right. I would propose slight modifications, though. Maybe re-word it into:
> City councils legally prohibit landowners from adding dwelling units to their own property.
Remember - all this "pent up" demand for housing is a result of homeowners being denied permission to modify their own property.
Historically, all cities evolved from fields because landowners incrementally upgraded their own property, when the price of rents justified additional improvements.
In the 1920s, city councils gave themselves permission to prevent property modifications "for planning purposes", and the US Supreme Court never struck it down.
The early goals of zoning were explicitly laid out by the mayor of Atlanta in 1922[0], with gems like:
Zoning will keep the apartments out of the private house sections. The coming of the apartment drives out the private home.
And:
The residence districts are further subdivided into three race districts:
R1 or white residence district.
R2 or colored residence district.
R3 or undetermined race district.
The above race zoning is essential in the in interest of the public peace, order and security and will promote the welfare and prosperity of both the white and colored race. Care has been taken to prevent discrimination and to provide adequate space for the expansion of the housing areas of each race without encroaching on the areas now occupied by the other.
Nearly every US city still has R1, R2, and R3 zoning, though of course no one includes the racial element anymore.
I gave a high-level overview/rant on zoning, racism, and Denver here: https://twitter.com/josh_works/status/1294726871574179840
[0]: https://josh.works/full-copy-of-1922-atlanta-zone-plan