Of course in reality it's not a private road or private sewer system or private fire service or private police force, it's all provided by government via taxes.
Taxes that people pay based on how much they earn, a far more obvious source of people being penalised for doing the right thing.
If you have an empty lot and you improve the district you can get your reward by selling the plot or opening your own business on it. Under LVT what you can't do is just sit on it for years at no cost to yourself, but great costs to others just in case it turns out to be a lottery ticket.
One assumes that the people I'm renting the lots to are permitted to use the road as part of their contract. But let's make this simple and say that I offer a perpetual agreement to let anyone use this road at cost, whether they rent from me or not. The road is a loss-leader; I intend to make my money by renting out the land, not by operating a toll road.
There are plenty of examples of communities with their own private roads and other services. Not everything of value to a community is provided by the government.
> you can get your reward by selling the plot
That won't work. Whoever buys it would be taking over the LVT payments, so that expense will come out of the resale value of the property.
> Under LVT what you can't do is just sit on it for years at no cost to yourself, but great costs to others just in case it turns out to be a lottery ticket.
And here we once again have someone failing to see the economic value of speculation.
The only reason to "sit on" the land and not use it would be that any use you could put the land to now would interfere with taking advantage of an even more valuable opportunity which you anticipate in the future. Not holding the land in readiness for that more valuable future use under such circumstances would be destructive.
Let's look at a hypothetical case for a moment. Say I buy some otherwise nice property for cheap because it's in the middle of nowhere and there is no easy way to get to it. Trees, lakes, etc. but no nearby roads. The LVT would be low, right? If you auctioned it off in that state as unimproved property you wouldn't get many takers. But say I build a private road to that area at my own expense, and then rent out the now-accessible land in family-size lots. The land hasn't changed, but its value is much higher now because of something I did. How would the "unimproved" value of those lots be assessed, and who receives the LVT? If the answer to that last question isn't "me" (and it won't be) then someone is receiving an unearned benefit from my work.
Even if you think it's somehow "unfair" for property owners to benefit financially from general improvements in their communities, LVT doesn't change the fact that someone is still getting an unearned external benefit. It takes from the property owner without doing much at all to ensure that the LVT goes to the parties actually responsible for the improvement.