Taxes aren't governed by a market; the same rules don't apply. Arguments against government protectionism in markets don't also apply to protection against the actions of the government itself.
That said, proposition 13 doesn't seem like a good implementation of this. There's no good reason for a sudden increase upon sale; that breaks the ability to buy a home. There should be a hard cap on property taxes that doesn't change on sale.
Why should taxes get to increase without bound or control on existing property? Once you've paid off your home, you should not have an ever-growing expense to keep it.
(You shouldn't have an expense to keep it at all, but that's a separate argument.)
My view is that property taxes fund the very things that make a particular neighborhood desirable, like good schools, roads, parks, and other local services. Your property's value is what it is because of these things, and you shouldn't be able to reap these benefits without paying taxes proportional to the value you've captured.
In California, that's less true than it might otherwise be, given the prop 13 limits on both assessments and tax rates. Lots of those things are funded by sales tax revenue, income tax revenue (primarily state income tax funding local programs), development fees, and other revenue sources other than property tax.
I think you're arguing that richer neighborhoods should have nicer schools, roads and parks. Granted, SF is an exception here, but that's precisely how things work elsewhere in California.
And if richer neighborhoods don't get that, California made annexations pretty easy, which is why you see those tiny municipalities in the vicinity of Los Angeles, San Diego, Anaheim, etc.
One can't really arbitrage that effectively.
Municipalities can't run consistent deficits, so the new schools, roads and parks will be built after a wave of newcomers buys properties, locking in higher prices and higher tax base.
In case there are no such newcomers (i.e. everybody is maximizing their Prop 13 benefit by not selling), the municipality just sticks to the last year's budget with a 2% increase permitted by Prop 13. But in that case new schools, parks and roads that make the neighborhood better don't appear either.
That's a really good reason to stop funding those things through property taxes.
Does it apply to anything else you own? If 2004 Honda Civics suddenly got really popular and I already owned one, I wouldn't suddenly be priced out. Or if I own gold or stocks, and the price goes up, that doesn't affect me at all unless I sell.
In California, since the state has ad valorem taxation on vehicles (the vehicle license fee, which is 1.15% of the market value of the vehicle) which does not have prop. 13 style limits, you, in fact, could be priced out of your 2004 Honda Civic if the market price suddenly and radically increased.
Prop 13 is a blatant wealth transfer between people who are new to the area (mostly the young) to the people who were originally here (mostly the old).