They could, but unless the state has any folks realizing property values increase over time and have the amount adjust accordingly it'll be broken in years to come.
Take Florida for example. First $25,000 of your property value is tax free if you own and occupy the property (homestead). This equates to about a $500 tax savings in most areas. In 1992 they added Save Our Homes, which limits how much of an increase the taxable value can increase each year (3% or the Consumer Price Index, whichever is lower). It works out great until you move or sell your house. Note that this only caps market changes in value to your property. If you make a physical change (swimming pool) the full value of that is added onto your property in the year you build it.
A few years back they added a 2nd $25,000 exemption, but because of the impact it only applies to non-school related taxes and only exempts homes over $50,000. This adds maybe about $300 in savings.
So why do I say this is broken? Back in the late 90s and early 2000s most homes were only worth around 50-80k, meaning this exemption knocked off 1/2 - 1/3 of the property value.. Now these neighborhoods cost $200k and up and the exemption barely covers 1/4. Granted there is no state income tax, but the average homeowner is going to pay around $2000-$3000 in annual property taxes. My current home is taxed on about $160,000 of value even though it is worth $280,000. Whenever I sell the new owner will be in for sticker shock at their $6000 tax bill since the cap resets on a sale.
Really sucks if you can't claim Homestead. With no exemption my 1200 sq/ft rental home 50 miles outside of Tampa costs $3000 in taxes a year. That means that $250 of my tenants' rent each month is just for the property taxes. Since there is no cap, if values jump up 10% that means taxes do too. Because incomes are so low I can't even follow the 1% rule (1% of property value as a guide for monthly rent). I'm closer to 0.7% otherwise I wouldn't be able to find tenants.