That is so wrongheaded I don't know where to start. Increasing property taxes is the exact reason you have older folks be forced out of their homes after having lived there most of their life.
No one should have a continuously growing tax burden over time.
Property taxes (specifically on the unimproved value of the land) offers a way out of this dilemma. If you can do more valuable economic activity with a piece of land, you can and should pay more for it, so it'll be worth more. So if you spend $10MM/yr in financing for building a bridge, and it increases land value in the catchment area by $100MM, and you have a 12% land-value tax, the state responsible makes more money.
The old person will make a good amount of money selling their home, and they can use that windfall to buy a nice new home somewhere warm and cheap, so it isn't like they're being trampled on.
It deconstructs communities.
This land is too valuable to allow you people to squat on it. We'll compensate you, so it's all fine and fair, now get out.
It's not hard to understand why people don't like being treated that way.
Subsidize? I think your concept of private property rights is under developed.
This is the gigantic pitfall of overly-individualistic property rights - that socially beneficial infrastructure development doesn't happen, because nobody has the power to siphon off enough of the value created to make it worth doing.
Prop 13 prevents this, and in fact makes it so that only redevelopment or change of ownership allows increases in property value to be taxed, which is one reason owners might prefer not to redevelop.
Full value assessment (even retaining the Prop 13 limit on tax rates) would encourage redevelopment because underdeveloped properties would still be paying taxes based on value increases from potential uses.
Like so what? Your property value, whether house or some artistic drawing, can go up or down based on other peoples actions. There is nothing wrong with that, that is how ownership of anything works.
When people loose business or work because those around them abandoned area or supermarket appears, they get no reimbursement. It is market force in practice. Therefore, when the same people earn on others coming in, they should not be punished either.
I don't know the English terms here, but that kind of situation where individuals are earning massive profits because everybody else built city around them, is just wrong. That unearned rise in the property value should be taxed away.
So, just zone more apartments in that area and put big enough property tax on underdeveloped land area. That way the economic incentive of the property owner aligns with the interest of the whole city.