In London this is through the Green Belt, torturous planning laws, height and density restrictions, low interest rates, government-backed mortgages and lending schemes, etc.
Supply in the markets for food, computers, software, etc. is not nearly so controlled by government action (or inaction).
Obviously the better solution is to do away with the restrictions, but politics is the devil's day job and all that.
Good thing there's no actual people involved.
For instance, imposing a very high land value tax would reduce the value of land, which would in turn reduce the cost of housing.
It would only potentially reduce the tax-exclusive cost of housing. It would not reduce the tax-inclusive cost. (In fact, the rational expectation would be that it would radically increase the tax-inclusive cost.)
You are right, that the tax-inclusive cost of housing won't change with a land tax on its own, at least not as a first order effect.
If however, you eg use the land tax proceeds to lower taxes on labour and capital, building higher (ie using more capital for housing) will become more economically feasible. (No economic revenue will be lost. Lower taxes in a jurisdiction directly lead to higher land prices---which the land tax will capture.)
There's also some other second order effects, like less NIMBYism. (https://www.dartmouth.edu/~wfischel/Papers/00-04.PDF) This will make development easier.
But I guess it's because of different attitudes in these two parts of the world? American don't trust poor people with money?