But does it matter? The way housing problems were always fixed was very simple and without any kind of tax: with the government building social housing, allocated according to social principles (you pay the same, but the size of the apartment is determined by the number of kids you have). A LOT of non-luxury housing. This has nearly stopped. These buildings were famously bad. First, they tended to be unfair and violate building codes, they were government projects (ie. cost a lot more than a luxury housing development despite being much worse), and since houses got allocated by the government to vulnerable groups, they tended to have safety issues in the first few years they existed.

But they fixed housing. A decade or so after they were built and things became somewhat stable around them.

The problem with tax, and any kind of economic system is very simple. Capitalism and communism, and the more creative options like mercantilism or militarism, we can argue about them for a long time, but fundamentally they divide up existing properties. Nothing more. We can use these systems to choose WHO is homeless, but they cannot lower the number of homeless.