The notion that housing prices are being driven up by restrictive zoning ordinances is true for expensive coastal cities like New York and San Francisco, which represent a huge minority of the housing in the United States.
The above linked Vox article refers to another Vox article to back up its general claim that "Housing undersupply is the result of decades of locals opposing new home building."
https://www.vox.com/22264268/covid-19-housing-insecurity-hou...
Then, this referenced Vox article references the following study as if it applies to all housing in general:
"In 2017, Yale Law professor David Schleicher wrote a paper called “Stuck! The Law and Economics of Residential Stagnation.” In it, he documents local restrictions on housing development, arguing they have become so overbearing that the increase in the cost of housing and rents has made moving to a better place impossible for millions of Americans. Local zoning regulations are strangling opportunity."
https://www.yalelawjournal.org/article/stuck-the-law-and-eco...
This study however is clear that the impact of restrictive zoning policies is localized particularly to coastal metro regions beginning in the 1970's:
"Something dramatic happened to land-use regulation in the 1970s and 1980s: it became much, much stricter. Importantly, while this phenomenon affected all types of municipalities—from urban downtowns to inner-ring suburbs to exurbs—it only occurred in particular regions of the country. In particular, coastal metropolitan regions like San Francisco, New York, and Boston restricted construction in cities, suburbs, and exurbs. Because these popular regions restricted new housing, demand for living space outpaced supply. Housing prices soared, but population growth did not.
In contrast to these coastal regions, Southern and Southwestern metropolitan areas like Houston, Phoenix, and Atlanta continued to impose minimal land-use restrictions. Though demand to live in these regions grew as well, this demand led to increased housing construction and population, rather than substantially higher housing prices."
Most of the recent growth and migration in population has been occurring in these Southern belt areas, which have limited restrictions on building. Housing prices is a demand driven problem, not a supply problem throughout most of the United States.