So if a LVT has the explicit goal of eliminating things like parking lots and self storage units because those don't generate enough income to pay for the taxes, then what hope do things like playgrounds and parks have to continue existing.. they generate far less income than a self storage facility.
Explain how.. In a dense urban area, with LVT, that lot that held a park will bring even larger tax revenue when the city sells it off to a developer. Having the tax be based on maximum potential usage will only increase the temptation to sell it off and remove yet another park from the people.
That creates a virtuous cycle for the local government who is administering those taxpayer paid amenities, same as other form of infrastructure and amenities.
Also, "luxury" housing cause what economists called "filtering", in which new construction are occupied by the upper strata of income, which means they pay for the cost. As housing age, this naturally becomes more affordable to the lower strata. This of course, depend on sufficient housing stock. Otherwise the inverse will happen.
Also, you only need to cover the cost of paying the land value tax to keep it, not to generate the maximum amount of revenue for that plot of land.
We are not talking about value extraction here, but making sure that landowners work for their keep, while the unearned income/economic rent that would otherwise goes to them is returned to society, because the value of the land is largely determined by the agglomeration effect, the sum total of the community's effort and entrepreneurial spirit. Otherwise, your private effort as individuals would flow to landowners reaping the benefit of increased land value, hence appreciation in real estate price.