If you bring more families into an area, the land around them may become more valuable, but there’s no reason that effect would be localised to the apartment block.

This is the core distinction between LVT and property tax.

That makes no sense. If land affects the value of the land around it then it had to have gone up.

If you put 10 family dwellings on land, then its value is at least that of 10 family dwellings. So each individual renter is somehow paying 1/10th the 10x greater cost. You haven't reduced the LVT by sharing it between the LVT between the renters, they're all paying the same or more.

If you buy a block of land with 10 filled apartments on it, and then you demolish it and leave it vacant, the land still has value.

In theory, it still has whatever value the optimal use would provide.

In reality we don’t have an optimal land use oracle, and valuation techniques would take the vacant lot as a data point suggesting that the area is bad. The assessed land value might decrease a tiny bit – along with that of your entire suburb, and similar regions. But it’s not going to zero.

Similarly, if you take a block of land, put 10 apartments on there, and fill them with renters, you’ll create property value where there was none, but the land value isn’t going to go up 10x.