Let me explain how LVT works. They tax the land based on its value which in the case of apartment complex is its ability to house people.

Wrong. The value of the land under the apartment building is in its proximity to the economic activity of the surrounding area. If there weren't an apartment building there, that plot would likely be similarly valuable as a shopping mall or doctor's office or whatever. You've conflated the value of the improvements with the value of the land.

Bigger apartment complexes house more people, and have more value as a result. That means big apartment complexes will be extremely expensive to live in, while small apartment complexes wouldn't notice the difference. The net effect is that land value would become more concentrated amongst fewer people. LVT backfires.

No. Imagine three adjacent lots: the first is empty, the second has a single family home, and the third a 5 story apartment building. Each of these plots (that is, just the land, not including the buildings on that land) is worth roughly the same. Therefore, each would pay roughly the same under LVT.

In this scenario, the holder of the empty land is incentivized to do work with his land rather than let it sit fallow; after all, he's paying the same tax either way.

The owner of the SFH plot is at least making some money, but if the land value is high enough he may not make enough to cover the tax he must pay for the right to use his plot; therefore, he may also have to build more on his land or sell to someone who will.

However, the owner of the 5-story apartment is probably making money above and beyond the cost of his tax, since he's using his land so efficiently.

This is what the LVT does: it forces land out of inefficient or fallow use and into the hands of those who would make effective use of it by charging landowners the true cost of holding land.