The part you're ignoring about prop 13 is how much it incentivizes nimbyism. If rising property values were accompanied by resultant higher property taxes, homeowners would be incentivized to work with renters to get new housing built rather than working against those that need a healthy housing market. But as it stands, prop 13 means the more they can block new construction, the higher the value of their homes.
> If rising property values were accompanied by resultant higher property taxes, homeowners would be incentivized to work with renters to get new housing built
I don't think the math works out.
Let's say your house is $500k, assessed at $500k, taxed at 2% (which is a pretty high rate; national average is more like 1.2%, and of course, California caps it at 1%). If you can finangle a 10% increase in property value per year, then in around 8 years your property will more than double in price, and you can cash in on $570k of additional equity. Meanwhile, you'll have paid an extra $46k in taxes.
Conversely under a vaguely prop 13 like regime (1% property taxes, 2% appreciation per year), you'll pay an extra $7.2k in taxes. That's obviously much better; you pay 1.3% of your gains in extra taxes instead of 8%. But...
...we're still only talking about 8%. Of the total value. Weighted towards the end of the period. On the margins a few people may be dissuaded from pursuing a maximal house price support strategy, but 8% is not a lot; if you own a large, massively appreciating, highly leveraged asset, most people can happily find the cash flow to cover the tax bill when it's that tiny.
In short: On an asset like a house, the capital appreciation overwhelms the property taxes.
(Also don't forget that the tax base can be re-assessed on sale. That part of Prop 13 actually suppresses house prices and discourages people flipping houses to profit from rising values, although I think that, like all the impacts of prop 13 on the housing crisis, it's very minor.)
The math only works like that if you realize the gain from the appreciation of your home's value. But on a primary residence, people don't want to sell. Selling, while realizing the gain, means either sinking most, if not all, of that gain back into another house, renting or moving out of the area. So for someone who doesn't sell their house, increasing property taxes can become problematic.
From your example, paying an extra $46k over 8 years per $500k of home value is much more significant when you're paying for it with income from working rather than an increase in your home's value. Nimbyism only works because you can both remain a homeowner and not have a significant increase in property taxes. You stop being a nimby when you sell your back yard. Higher property taxes would force homeowners to choose between their objections to new building, which increases the value of their home in some distant future sale, and increased property taxes in the immediate future. The time horizons are different, which makes the kind of comparison you're making problematic.
