There are many vacant houses in the U.S., and many cheap houses -- pretending this is about shortage of physical housing is just foolish.
This is about shortage of desirable housing, and instead of stopping to ask what are the mechanisms that cause desirable housing (for you) to be so expensive, you just assume the problem is like a shortage of lumber or "space" or something equally irrelevant. If you build lots of affordable housing in Livermore, what would happen is that Livermore would stop being desirable for you, and you would end up wanting to live in some satellite area that was created outside of Livermore in which the exclusivity was maintained. Then good jobs would move there, because over the long term, the jobs follow the skilled workers and not the other way around. Then your utopian project in Livermore would find itself turned into a low income area where there is a shortage of good jobs, and most of the good jobs are elsewhere, in more exclusive areas, that are within commuting distance of the Lab. Then the Lab would find itself having trouble hiring enough people, and it would open satellite offices elsewhere. Then you would complain about a shortage of housing there, and want to rinse and repeat, not understanding why your intervention failed to solve the problem.
A good example is San Francisco. Most tech jobs were in Santa Clara county, not San Francisco. But a lot of tech workers enjoyed living in SF and did a long commute. Then tech companies realized they could hire more easily if they opened offices in San Francisco. Then companies began to be founded there, etc. All of a sudden, lots of tech jobs in SF. The jobs follow the skilled workers, and skilled workers have a lot of options in terms of picking where they want to live. They vote with their feet, and the jobs follow. How did the jobs end up in Santa Clara? Or Belmont. Through a similar process. Jobs chase high skilled people.
So you have to come to grips with why people that have a lot of choices choose to live in expensive, more exclusive areas. Why does the average high income person really not want to live in a neighborhood with a lot of poor people? Or, for that matter, even average income person? Why this hatred towards average people (something that can easily be found even on this site)? Why are they willing to commute elsewhere, and even take a pay cut, to be in a more exclusive area that is surrounded by other high income people? (And so signals to employers to open an office nearby). This is a social issue, not about lack of nails or space or a shortage of lumber.
I do have some arguments in favor of your position, but they’re mostly specific to the US: 1. Many people’s primary retirement plan and emergency fund is often their house, so property values are exceptionally important to them. 2. School funding per student goes down if less wealthy parents move to the district. 3. Violence in poor neighbourhoods can get really bad. None of this is God-given, though, which is why nimbyism isn’t quite as bad in many other countries. And while important, these factors don’t seem to be dominant.
The fact is, there just aren't that many cities that fit this bill, and those that do experience rapidly rising real estate prices. As with many issues, race is a taboo factor operating behind the scenes in determining which neighborhoods are targets of gentrification and which are not. When you create a lot of low income housing, people viewed as more undesirable move in and the high income earners go elsewhere. Then you rinse and repeat. But of course the problem is not that there's a shortage of housing or even a shortage of affordable housing. Median value of owner occupied units is $245K, which is 3.5x the median household income[1]. That's perfectly affordable. But the median home value in a major urban city with less than a 10% African American population is a different story. So there's a shortage of housing in our diverse society that has the "right" kind of demographics. As we become more diverse, the number of locations that are predominantly white or asian will get more and more scarce -- and more and more expensive.
You see a similar situation in Sweden, which accepted a large number of migrants and put them on the outskirts of major cities, and of course the house prices of units in the urban cores that were free of migrants skyrocketed. People are paying to not live in areas with large migrant populations. So all of a sudden housing is becoming 'unaffordable' in Sweden. But the unspoken part is that housing with few migrants is become unaffordable, not all housing. Again, this is a social problem, it's not a problem of a shortage of lumber or space.
So building more low income housing doesn't have the effects you believe, in terms of increasing the amount of housing in highly desirable areas. It will have the opposite effect, by removing one area that has the desired demographics, it makes the remaining islands even more desirable and thus even more expensive. At least this is my reading of history. It was very clear, after the race riots of 2020, how home prices in predominantly white/asian enclaves began to rapidly increase, at which point those who wanted to live in these enclaves began complaining about a "housing crisis" because they could not bring themselves to complain about a racial crisis.
But hey, I could be wrong -- housing in St. Louis or Detroit is still affordable. I'll wait to see tech workers scramble to move there, rather than, say Salt Lake City.
[1] https://www.census.gov/quickfacts/fact/table/US/VET605221
