First problem: This is a recurring process but the land is only "unimproved" at the time of the first auction. At which point no one was interested in the land, there being nothing and noone there, so the auction price is effectively zero. After that first auction you can't say how much of the market price in any later auction is due to the land itself and how much is due to improvements.
Second problem: In order to auction something off you have to own it first, but this is unowned land. No one has the right to sell it—or to prevent anyone else from using it. Land only becomes property after it has been improved. This is also the point where it first acquires economic value.
Third problem: You can't just go auctioning off already-improved land separate from the improvements. The improvements and the land are a single, unified good and in the vast majority of cases cannot be separated without incurring expense akin to rebuilding the improvements from scratch.
Fourth problem: LVT is valued mainly for being a tax on "unearned" external benefits property owners receive from the community, but the "unimproved" value of the property is a poor proxy for net external benefit received by property owners. On the one hand, the value of the improvements also depends on the community; if you build a family home or a restaurant or a warehouse the value of that building is going to depend on the demand for specific kinds of goods in that area and not just the cost to build a similar building. On the other hand, the so-called "unimproved" value of the land is readily influenced by the actions of the property owner, for good or ill. LVT punishes property owners who contribute to the surrounding community by taxing them on any increase in the value of their property which follows from their own contribution to the public good, and rewards property owners who contribute to lowering property values in their own neighborhood.
To your second problem: Where in the US is there unowned land? The settlers went west across thousands of miles of unoccupied land. Why didn't they stop and settle? It was owned. Early and ultimate sprawl. Others had pre-empted it for the benefit of their heirs. Think how the development of the US would have been had development proceeded west in some orderly way.
To your third problem: the land is forever. If well taken care of, it will remain fertile. But its fertility may not be important for all that long, depending on its location; its highest and best use will stop being agriculture. Buildings built for one purpose gradually deteriorate, and as the community changes, are no longer suitable to the current highest and best use of that land. At one time in NYC - say, 1880 - 21 year leases were common, and huge department stores and other edifices were built on them. It was understood that another building would likely follow. And it did. And the landlord got his "share." How did he earn that share? And look at the teardowns in your part of the country. Homes that seem perfectly serviceable to some of us are bought to be torn down to build something else on that well-located land. It doesn't happen on the edge of town; it happens in the center of things. The price one is willing to pay to get that land is pretty clear: the transaction plus the cost of removing the old improvement.
To your fourth problem: the land owner who doesn't improve his land can still sell it for a higher price due to the activity of the community around him. The value of the improvements never exceeds their replacement value, but could be quite a bit less. A huge spa resort built 15 or 20 miles from Disney World is likely an over-development of the land it is built on, while closer to DW, it might be quite appropriate and profitable. One who is building with consideration to others' preferences -- as customers or as tenants -- will build differently from one who simply wants what they want, without regard to resale value. We don't increase the unimproved value of our own sites, but we can contribute to the value of the surrounding sites -- or we might decrease the value. Increasing it doesn't exempt us from any portion of the value of our own site. Our neighbors' actions can increase or decrease the value of our own site. Regular reassessments take that into account.
The authors seem not to understand the topic unfortunately and have to use long prose as they are trying to wrap their head around it.
The core problem and idea for solution is actually quite simple. Let me condense the topic into a tweet or two.
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Problem: You have a naturally occuring resource which is scarce and which nobody created. How do you allocate it to humans?
Solution: Georgist simply say you can use an auction granting you possession for some amount of time instead of giving you time unlimited rights just because you were first to ask to use it.
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This is not a theory as it is applied in many systems such as DNS allocation or electromagnetic spectrum allocation. Georgists point out that similar allocation problem is for "land" (think location) and the same auction mechanism can be applied.
Land was considered abundant in the past, especially as new continents were being discovered. Therefore nobody though about allocation, same as nobody considers allocation of air. If air got severely polluted (say by radioactivity) and only small fraction remained usable for humanity, you can be assured that the same allocation problem would exist for air.
After discovering all continents on Earth land ceased to be abundant and now requires a proper allocation mechanism. Otherwise landlord monopolies will continue piling up profits because of their luck in winning the infinitely durable possession rights allocated years, decades and centuries ago in badly structured auction.
Either we start inhabiting new planets or we solve problem of allocation of important scarce resources.
What's funny - you solve this allocation problem and many societal problems disappear as if by magic. Somehow this suggests that land allocation is a major root cause of society.
That's why Georgists dwell on it so much. You should start solving problems from the most serious ones with high impact.