Decrease property taxes and prices will immediately increase by the previously-taxed amount. If the bank was willing to give $X loan with property taxes included they will still give the same loan with property taxes excluded so the market will just adjust higher.
Even if the total cost of a mortgage inflates to the same amount one would pay with property taxes in place, the mortgage payment doesn't fluctuate with property value and stops when it's paid off.
While those who have nearly or fully paid off their mortgage (which admittedly does disproportionately select for old boomers) before abolishing property tax would benefit most, more predictable pricing, a rent-free end goal, and much less risk of being taxed out of your own property are still juicy benefits for anyone who will ever own a home.
Might also be able to get rid of government property assessor jobs which could save a bit of tax money and mitigate a bit of the inevitable hike on other tax sources.
Personally I think if property taxes were abolished there's a strong argument for it to only be on primary residences, as I think an incentive against scooping up multiple homes, particularly as financial vehicles, is a valuable thing in society. No need to price landlords out of existence, but there should be some friction that favors homeowners over landlords.
Making the taxes conditional does mean we can't get rid of assessors, but the difference in income and effect on society may be worth it and still means assessors won't be bothering normal people.
