On a personal level, I like the LVT as a single tax because then I wouldn't have to do any paperwork. They give me a bill, I pay a bill, there's no exceptions, I can improve (or fuck up) my property, and overall the numbers won't change (unless a ton of people improve or fuck up their properties around me as well). Nice and easy.
But it also seems effective overall. After I quit my last job and before I got my current job, I did not work for several years (some due to choice, some due to circumstance). During this time, I obviously paid no income tax. But I still paid my mortgage (some of which obviously went to property tax, but mostly to the bank). Presumably with a land value tax, my payment to my mortgage company would be notably less less and the tax would notably be more, so during that time instead of paying a bank I would be paying taxes that go to infrastructure and welfare and whatnot.
I think this sort of thing is more accurate than an income-based source of taxes, because I obviously wasn't really in the same economic situation as an underemployed person who truly couldn't find a job for several years. It also allows you to assess your own level of taxation. For instance, I could have sold my place or rented it out, and moved elsewhere to a cheaper place (or I guess a more expensive place, were I so inclined to do so while not having a job).
Of course, the big disadvantage, as mentioned elsewhere in this thread, is that it is difficult to assess land value sometimes. It also seems like it may force people to move a bit more often, and moving is exhausting. And also presumably you'd have to convince people to pair this form of taxation with more lax zoning laws (with some stricter exception for farmland or something to keep the value per square foot low enough for it to still make sense).
Even weighing these factors, it seems like an improvement over the current system. But I guess the real issue is that in the current world of mortgages and such, even if 90% of the population decided it truly did make sense as a form of tax, the transition seems nearly impossible.