Take a look again at the graphs in the article the GP posted[0]. Texas's tax scheme is about as regressive as it gets. Sales and property taxes are regressive. Income tax with marginal tax brackets is progressive. This is a pretty simple concept.
> However, when property taxes are higher, there is more incentive for cities and households to allow more housing.
True, and what tends to happen in CA is that cities will approve commercial development over residential, since they can get much more tax revenue out of commercial property.
But I don't think CA's property taxes are too low; from a little searching it seems like CA's average property tax rate is only slightly below the national average. Meanwhile, Texas' clocks in at several multiples of the national average.
I'm not really sure what you're getting on about with the rest of your post, though. Property taxes, as implemented pretty much everywhere in the US, are regressive. Full stop. It doesn't matter "who writes the check". Landlords will pass on the cost of property taxes to renters. I don't really see evidence that home prices are so much lower in Texas due to the high property tax. Lower, certainly, but I would still much rather buy a house in the bay area with its inflated prices over a house in Austin.
And I expect as demand goes up in Austin, home prices will go up (if slower), and tax rates will just help make homes even more unaffordable there for lower income folks. I expect it's easier to build in Austin, but it still takes time, and likely any new demand for housing in Austin isn't primarily in the outskirts of town where things are cheaper.
[0] https://www.motherjones.com/kevin-drum/2019/11/taxes-are-sur...