My city assesses land and structure value separately as part of property taxes so having them assess just land instead seems like less work not more.

Mine does that too, but the process is clearly flawed because you can't buy land for the assessed price, and you can't build a structure on that land for the assessed value.

Just because it’s not possible to separately buy the structure or the land doesn’t mean the numbers are wrong. The point still stands that taxing land disincentivizes nothing while taxing structures discourages building them. Many towns in Pennsylvania had a pretty good track record with this. They implemented a mixed tax system that focused more taxes on land and less on structures because they wanted to make people sitting on empty land and not using it pay more. They also didn’t want to penalize people for upgrading their home. The system was extremely successful in its aims.

Rebuild cost and current value are different, given that new construction is worth a premium

It works because they just need to come up with a total value and how it divides between land and house doesn’t matter much. My property taxes also separate out land and it puts the land at like 25% of the total value, but in reality someone would easily pay >50% for my land just to demolish the house and rebuild.

These end up with wildly differing figures. Somehow my purchase price is $772k, my insured price is $225k, my property tax valuation is $302k, and the market price now is either $800k or $1200k, depending on if you're asking to sell it or use it as collateral! Nobody agrees on what property is worth.

Sidenote... you're almost certainly under-insured, especially if you're in California. If a fire sweeps through, you won't be able to rebuild much with $225k. Might want to check that out.

Insured price is based on the structure, not the land. If your house burns down, that's the part they need to spend money replacing. Valuing a structure's replacement cost is (fairly) straightforward based on materials, current labor costs, etc.

And so there is a way to get a reasonable approximation of land value: Subtract the insured value of the improvements (which should be accurate by virtue of being determined by a profitable insurance company in a competitive market) from the appraised property value (which there are generally accepted methods for figuring).

Sort of, but that would give you the "social value" of the land, which it only has because of its present use and its proximity to other land used for specific purposes. E.g., the land under my house is worth $X because it's in a residential area in a major metropolitan area. If I were to build a dense mixed use complex on a large plot of land, I would probably increase the land value (since density would make it a desirable area).

Does LVT look at a piece of land's productive value instead (i.e., how much food you could grow on it or how many minerals you could mine out of it)?

The point of an LVT is to tax all those external factors that increase the value of the land, such as local amenities, zoning, etc. The idea is the only value added by owners of the property was the structure they built, that’s theirs, the rest of the value is in a sense un-earned. Much of it is due to government investment in infrastructure and facilities, for example. Taxing the social value is the intention.

But if a developer builds, say, an airport on some unattractive land outside of a city, they will create most of the value being taxed, right? The same is true to some extent for any structure.

LVT is based on location, not farming potential.

It’s not 100% foolproof but just a rough guide because demolition and waste disposal costs are non-zero. For someone else to use the land my house sits on someone would need to deal with the asbestos likely in various walls. Additionally, things get tricky if the land is found to have historical / archeological relevance which can stop development indefinitely. Commercial developers carry an insurance policy for this I believe.

However, credit isn't approved every year so it goes out of date which is why cities, usually yearly, do regular property value reassessments.