I'm a life long Californian. I dislike Prop 13 and it's evil sister Prop 58 which allows children to inherit their parents property tax break along with the property. AKA my dad has a house with an assessed property tax value of about $75k, when he dies and my sister inherits it, she'll also inherit the $75k property tax valuation as well.
That said I don't think the current issues with housing in the bay area have much to do with Prop 13. It's why your schools suck. It's why the prices are high, because the carrying cost is 1% lower than you'd expect because property taxes are too low.
Real issue is the deranged FIRE based economy we have. FIRE Finance, Insurance, REal estate. Combined with low growth and income inequality. Means you have a lot of money pouring into 'hot markets' like SF, Boston, Vancouver etc. Same time there is collusion between the people that sit on the boards of large companies to suppress wages.