It's not about raising more money, it's about fairly apportioning the tax burden. Why should two homeowners with the same value of property pay different amounts of taxes just because one is old and has owned the house a long time?
It's not like income tax where your tax bill only goes up when you get more money to pay it with.
If the value of your house goes up because you remodel the kitchen, then it seems fair that you should be able to pocket that value. You did the work, so your taxes should not increase.
If the value of your house goes up because the government opens a new light rail station nearby, then you did nothing to earn this, so it seems fair that society should be able to get higher taxes from you.
In a speculative bubble, you did nothing to earn the extra value, but---arguably!---neither did the government, so it is less clear cut.
I assert it's a problem with a non-trivial solution.
If you have a house built in 1902 in the middle of a big city, you're paying minimal taxes and find yourself surrounded by high-rises, it's very inefficient land use. Your lot could provide housing for hundreds if not thousands of people if another high-rise was built there.
Thus, you should either pay for your privilege (in taxes) or sell the lot (for a lot of money, presumably) to let other people use the space more efficiently.
However, I must say that a property tax itself is a wicked one and should be replaced by land value tax. Land is more valuable in the centre than in the outskirts of the city, so when properly taxed it creates the tendency to increase density of the highly taxed land and allows for more liberal land use where nobody else wants to build.
In contrast, land value goes up in the average if a lot of people keep building desirable buildings in what turns out to be a desirable district in the city. That promotes what people like. Consequently, inefficient land uses move further away from the desirable areas.
Implementing neither property tax nor land-value tax is not straight-forward in general and there are corner-cases that need to be sorted, but in general land use is what ultimately provides its value. Often property tax codes already incorporate parts of land-value taxing, but the intrinsic value of property to be taxed is debatable: what is the value (to the community) of a small, low-taxed detached house in the downtown versus a huge, high-taxed mansion outside of the city? The opportunity cost of taxing property might be huge.
Since you want them to move out so bad, why don't you have the government pay them to move out? I never see these local governments offer to pony up $10k or so for the moving expenses for people priced out of their houses to move out. These people bought a house in a town thinking it'd be a good place to settle down, not that they'd be forced out of their house in a decade or two. It's really no better than forced relocation, which is a form of genocide.
Being priced out of a market is not the same thing as genocide.
Population transfer is a different thing from gentrification is a different thing from eminent domain is a different thing from genocide.