Also, housing has a lower price limit. Adding more housing is expensive. Building new housing will always be more expensive than, say, ten years ago. It will also be the more expensive the higher you build. This premium needs to be paid for by rent. Thus, new housing will never filter completely to the bottom, unless the owners go bankrupt.
Cities attract ever more people and that's the core reason of the housing crisis. Ironically, housing, and generally the lack of space, is one of the few factors that actually repulse migration into a city. Ease it and migration will increase, worsening the space (and housing) problem again, probably to a new level.
I would argue that it's pointless to try to "solve" that problem, as long as cities profit from ever more people living in them and people economically profit from moving into or close to cities. Any attempt to mitigate the problem will only lead to more migration into the cities, forming a vicious cycle.
One "solution" would of course be to let housing become completely unaffordable (e.g., just stop increasing or even actively decrease housing). Areas then become uninhabited by normal people. For certain areas this seems to be largely accepted: I don't think anyone has complained lately about the housing crisis in London's financial district.
Another option would be to make it completely unattractive to actually live in a city. One could, for instance, restrict all buildings to single-bedroom apartments by law, move schools to the outside if the city, etc. People might then spend the night there, but will hardly live there.
The interesting question should be: Why do people pay as much as they do today to live inside a big city? What makes it so much more attractive than living 5km further outside with 5-10 times the space per person?