It is mathematically impossible for someone in poverty to be unable to afford property taxes, because if their property valuation is so high that taxes are a burden, they're not poor.

For someone on Social Security, that home may be the only asset of any real worth they have. Social Security pays out an average of less than $2,000 a month. We can squabble over the technical definition of poverty, but look at the reality of it. A 70+ year old person on Social Security doesn't have good odds of getting hired anywhere that's gonna pay him worth a shit. They can't afford modern rent prices on that sort of check. Their only real shot at staying housed without a bunch of other retired and poverty stricken roommates is to have already paid off a home. Their financial situation is very likely to never significantly improve again for the rest of their lives.

Now, I'll admit some states have very low property taxes that won't impact things too heavily, but that's not universally true. Look at New Jersey. They have a property tax rate of 1.86%. For that to constitute half of the average Social Security check, as mentioned in OP, that'd only require a home with a value of $640k, which sounds like a whole lot until you realize the median NJ home price is $540k. That could be a fairly run of the mill house that used to be rural and got caught in urban sprawl, spiking the value. That could be a modest home on a very little bit, not a lot, of farmable land. That could be a home in a rundown part of town that got gentrified over the last decades. That could be a few critical companies moving into the area and spiking home demand. That could just be our housing market doing what it's done for the last half a decade and just belligerently raising prices to ludicrous levels.

I don't think that sounds like he's living it up. I think that on a $2,000/month budget, even if his home value excludes him from the technical definition of poverty, he's still gonna fucking feel like he's in poverty, especially if you fuck with his housing.

And yes, if the housing market happens to be whackadoodle and despite the sale proceeds they still can't afford rent for some reason, then they'd be eligible for subsidies.

Why not just leave them there in that case? What's the sense in forcing them out of their home just to push them into a new home that has almost the exact same problem? Now you're paying for subsidies and paying to manage the subsidy program instead of just... Not taxing them. It's counterproductive.

Including people whose homes, through no hard work of their own, have ballooned to incredible value.

Sure, but you seem to be drastically overestimating what it takes to get there. ALL home prices in America have ballooned to what should be considered incredible value, especially looking at modern build quality.

A person who becomes a millionaire through property value increase is even less deserving of tax breaks than a business owner who makes a million dollars.

And this is why I specifically said to cut the tax for reasonable homes. Dude in a McMansion can downsize. Dude in a slightly over average value home, though, can stay put and forego some taxes as far as I'm concerned. Set a threshold, but tie it to local property values. An average home should be fine. I might be willing to agree to double, but I'd have to think and research more. But beyond the value of a reasonable home, sure, levy taxes on the excess. Something like full property taxes on any value over some threshold.

At least the business owner probably put some work into earning the money.

Eh, I think business owners get too much credit. The vast majority of value created by all but the smallest companies is created by the workers. Most business owners depend on exploiting their workers. CEOs sure as HELL aren't working hundreds to thousands of times harder than their lowest paid employees. Someone that's self-employed, sure, busting their ass and earning it, but business owners on the whole, no.