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.
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.