At the surface it seems there should be a market solution for this.

One approach might be to do something clever with mortages and property rights. Basically keep trying to keep the current mechanics of borrowing and just shifting around the parties involved so that the all the interest gets collected as LVT.

Might even let the central bank handle both collection and dividends of it all.

Another option might be to create economic incentives for some party to get the assessment right. Like insurance companies and banks loose either customers or revenue when getting it wrong.

Perhaps can play with some margin for speculation such that lender and borrower get to share some profit.

Or just make lenders outbid each other on who can extract most rent for the property.

I guess one concern here is to make sure the commons have a representation, so that not all land get appropriated to private use

This is a 'clever' solution to property taxes. But a land value tax specifically just taxes based on the value of the land. This is supposed to incentivize people to use their land as productively as possible -- with a property tax, any improvements you make increases your tax bill, with a land value tax, it doesn't.

A moral argument for it is, if you own an empty lot which is gaining in value as the community grows around it, you are essentially leeching extra value from the community through no work of your own. This sort of behavior is punished heavily with an LVT, as your taxes go up but you don't get any extra revenue from your empty lot.

The problem is that most land has stuff on top of it. The government can't buy the land without the building that comes with it.