The best argument against LVT that I came across was from Bryan Caplan concerning prospecting for oil; once a person finds oil and builds a well, LVT would not charge her for the well, but it would charge her for the oil. She didn't make the oil, but finding it does take effort and you don't want to disincentive that.
I still like LVT conceptually but I don't know what the best solution is for this kind of case.
Of course, you could alternately just sell your now-valuable land. However, the higher the ongoing tax bill (due to the LVT) the less people will want to pay for your land. The LVT captures some or much of the value currently accruing to landlords (depending on how high it is).
High property taxes can serve a similar function. Taxing the unimproved value of the land removes a disincentive to improve the land, though. If you knock down your house and build a much better one on the same land, your property taxes would go up while LVT would not.
LVT vs property tax is not the main point, though, in my opinion. The real issue is moving from heavy reliance on income tax to heavy reliance on property tax or LVT.
Granted, any LVT won't approach the 100%-of-rents rate, so it's a proportional thing.
>for the purpose of encouraging building housing how is a land value tax functionally different from a property tax?
A land-value tax is indifferent towards a single-family home and a thirty-story apartment building on identical plots of land. Property taxes aren't, they add an additional penalty on the highly-developed land.
Shameless policy plug: part of the reason that property tends to be a good investment is that a significant portion of its value derives not from improvements, but from "location, location, location": proximity to businesses and community services. Some economists (Henry George, Milton Friedman) have advocated shifting property tax to be based on the unimproved value of land, rather than its total/improved value, thus offsetting the (arguably unearned) benefits of sitting on a lot while the surrounding community accrues value.
https://en.wikipedia.org/wiki/Land_value_tax
https://en.wikipedia.org/wiki/Georgism