Prop 13 is just a phenomenal transfer of wealth from the young and working class to the old and wealthy. Don’t kid yourself that it was ever about anything else.
Under Prop 13, the assessed value is X when you buy it and can only be increased by up to ~2% per year even if your home went up in value by much more. This chains together over long periods of time, so that if you bought a place 20 years ago you are now paying property tax on at most X * 1.02^20. If the new owner paid you 5X for the home, they now pay property tax against that value moving forward.