In my area, rents have gone up concurrent with house prices - but aren't falling now that sales inventory is falling, because those owners still want to cover the mortgage.
> property taxes should not be based on the "full cash value" of a property
That's not quite how property taxes are calculated, at least not in the two states where I sell real estate. In fact, even after a house has recently sold and there is a new market value established, the property tax basis doesn't increase up to that new market price. I just went through this with an out-of-area buyer, where I took the houses I sold last year that were comps to the one they wanted to buy and showed them the new tax assessment for this year. Yes each had increased, but nowhere near the market value (in fact, the tax assessed value was between 50-60% of the market prices).
Look at how rents are calculated - over a long period they usually are somewhere near costs, of which the largest costs by far is the mortgage, which depends on the mortgage rate. Property owners are often speculating on the capital gain. Commercial property is different again.
You also need to consider what behaviour that would incentivise.
Finally, the property value in some markets is entirely dependent on rents (especially commercial) so you are not necessarily gaining anything.
To make it less volatile they might as well just pick a number every year, the annual tax for a median house, then scale it in a relative sense based on relative house prices between taxpayers. They would more or less do that in a roundabout way anyway by tweaking the percentages every year.