For those unfamiliar, you can think of Prop. 13 as an effective cartel to discourage people from selling their property.
Basically, it subsidizes keeping property and never selling it. This is because every year the owner's property taxes essentially go down, presuming any normal level of inflation. And because of follow-on propositions, you can transfer that advantage from parents to children. Yes, this does let existing residents stay somewhere indefinitely. Equivalently, we can say it strongly discourages mobility.
The crazy real estate prices you see are simply a result of the proposition-established cartel. This drives up prices for new arrivals, which California depends on to keep this going, which makes the people holding on to their property think two things: "I'm rich because my home is worth so much" and "I could never get by without Prop. 13 because my home is worth so much." This has made Prop. 13 untouchable — it would basically have to be overturned at the state Supreme Court level at the behest of broad popular opinion.
However, this is just another bubble. California as a state has done relatively quite well in the last thirty years, so it continues to inflate. However, California and local governments have also spent a huge amount of future revenue on state workers, so it has become harder to sustain. But for the huge amount of income tax receipts from the investment class of California, things would already be in rough shape.
Someday, the bubble will burst. If Prop. 13 was ruled unconstitutional tomorrow, it would cause a real estate crisis, followed by an economic crisis, that is hard to imagine. But that might be preferable to having Prop. 13 crumble in the middle of a statewide economic crisis, which is the most likely ending of this story. In the meantime, yes, it does keep grandma in her tidily-appreciating house instead of some filthy tech hipsters chasing the next gold rush.