Property should be taxed at the price it was purchased, and the elected official should set the appropriate rate needed for the budget. This puts it all on the elected officials and the voters. They would know the minimum tax base valuation (assuming no property changed hands, and could plan accordingly.
The appraisal district should only be used to asses new values when property changes hands due to inheritance, gifting. or foreclosure. Basically, anytime the property changes hands without going through a sale. There might have to be periodic adjustments for commercial properties owned by corporations, as they could theoretical be owned in perpetuity, without changing hands.
Currently, the appraisal district sets a value, property owners protest it is too high (even though they would not consider selling at that value,) and elected officials are spared having to make major rate changes. The governing bodies get more money and elected officials boast of not raising rates.
***It's your money, not theIRS! (At least for a little while longer.)