https://www.top.org.nz/higher-incomes-policy this is the closest we have currently. If you're a pensioner you can defer payment until sale of the property, which is technically "unfair" in that non-pensioners miss out on access to that cashflow, but it isn't politically viable to kick old people out of their homes for non-payment of the tax.
I don't think that landowners are to blame for the problems we have, but you're not considering that about half of the population of NZ do not own their own home. They are currently being screwed over by the system, and landowners on the other hand are benefiting disproportionately. Resetting the system to be more fair does result in some people losing, yes, but equality tends to feel like oppression when you've been benefiting from inequality.
Of course, most people that would start paying the land value tax would get a reduction in income tax, and would also gain intangible benefits like their children and grandchildren being able to afford to live in the same city as them. Currently the government is stealing away large amounts of the profit of our labour (income tax), so we are reducing "communism" by allowing people to keep more of the fruit of their labour.
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I know it's a fairly extreme analogy and probably invoking Godwin's Law, but slaveholders in the UK and USA made similar arguments about losing their investment when they were just working within the system and slavery was in the process of being abolished. They worked and saved to purchase slaves, and then the government was proposing to simply destroy all that investment by freeing the slaves.
The UK government actually ended up compensating slaveowners for their loss of investment, which I think was wrong even if politically expedient.
The analogy works sufficiently if you can answer two questions: 1) why does land increase in value, and 2) is it fair for tenants to miss out of land value gains but landowners to get land value gains?