I certainly wouldn't say that Prop 13 is an unalloyed good, but you're partly misunderstanding how it works. The assessment only goes up- or only goes up by more than 2%/year- when the property changes ownership, i.e. when it's sold or inherited, not when it changes occupants. This means it has the kind of effect you're describing mostly for owner occupied housing, not for rental apartments. What it really does is to give a tax advantage to owners who have owned a long time, whether they live in the property or not.
It's also important not to exaggerate the importance of property tax in the overall cost of a property. Prop 13 also rolled back property taxes to 1% of the value of the property, and while there have been some tax increases since then, typical property tax rates in California are around 1.25%. If somebody wants to sell their house to move into a smaller house, a big jump in real estate prices will generally give them big windfall profits on the sale that will soften the blow of higher taxes on the new place. There's also a special clause that lets people over 65 carry some of the reduced assessment from their old house if they sell and buy a new place; value in their new house up to the sale price of their old house will be assessed at the assessment for their old house, so their taxes won't go up unless they're moving into a more expensive house.