"People would move more instead of being locked in to a low effective property tax." - this just means that older and retired people would get kicked out of stable living situations, driven by property tax assessments which are often driven by real estate market boom/bust cycles. And housing prices would be just as high as with prop 13.
The bad part about prop 13, is that it never should have applied to anything other than primary residences of people. Having it apply to rental & commercial properties is what warps the market.
Cities should be cutting their police budgets - most small towns have nearly 50% of the budgets tied up in police. All while there has been a multi-decade decrease in crime.