Whether this theorem is robust against systematic cognitive biases is another matter. I don't know the answer to that, but it seems like voters are more likely to eg. vote for a tax that falls on someone else but gets passed along to them as higher prices, which would be an argument against an LVT ever being enacted.
Also, elsewhere in these comments are good arguments why an LVT is very difficult to "bolt onto" and enact a government once it's already been formed, which may be why we don't have one. That's simple path-dependence: Georgism didn't exist when most modern industrial nations were first formed, therefore it can't exist now.
This, incidentally, is the function of propaganda. You introduce an influence that biases the system to encourage certain classes or error and discourage others, thus skewing the consensus-forming process.
The idea behind the theorem is that all of these public benefits - open space, parks, museums and cultural attractions, mass transit, transportation, safety, etc. - increase the relative attraction of living in a location. Hence, they raise the rents that landlords can charge. Places like NYC or the Bay Area are attractive in a large part because of public investments in social goods there. The increase in rent due to public goods is specifically the increase in land value - private improvements to an individual property are not reflected in the LVT, but that portion of the property's value that is common among all properties in the area is.
Therefore, every landowner has an incentive to pass a tax increase iff it will raise the rent they can collect (or the implicit rent, for an owner-occupied property) by greater than the amount of the tax. And they have an incentive to ensure that the tax is spent on benefits that actually improve the appeal of living in a place, because otherwise they're losing money on the deal. Hence the theory implies that a LVT is not only the optimal way of collecting revenue for a government, but also leads to an optimal size of government, making it popular among many left-libertarian types.
[1] https://en.wikipedia.org/wiki/Henry_George_theorem