That is the achilles heel of Prop 13. In it's effort to keep people from being taxed out of their homes ( particularly for retirees who may not have the resources to stay on at the higher rates), the law essentially slowed the turnover in places where incomes are lower. Here where we live, home are quite expensive and incomes of residents, and future residents are high enough that there isn't a big problem with affordability. As an example, we build our home in 1983. Home values in the area are now around $ 2million. The home across the road just sold for $2.1million. The taxes the former owner paid on the home for which he paid something on the order of $1.5 million, were in the neighborhood of $18,000 per year. OTOH, our taxes are around $7,000 since we've been here for more than 30 years. The real problem is that the property tax, should be relegated to paying just for property-related services, but today they are mostly comprised of taxes to support public schools and county government, but making the needed changes are too hard for today's legislators. They are happier legislating plastic grocery bags out of existence, messing with gun laws and other de minimus acts where courage isn't required.
I think a big problem is that when you sell your home, the prospective buyer will face much higher taxes.