I think Austin has a decent chance.
1) No prop 13 which keeps locks California property taxes at purchase price. It means long-term residents pay very little tax on very expensive property, so are unwillingly to sell (better to cash out with a refi and let the property keep appreciating, which creating real estate scarcity, pushing up prices, etc...
2) Austinites are well aware of the failings of California cities such, and fearful of making the same mistakes. Even the left leaning voters that flee to Austin from deep blue cities are by and large not the same group of voters that push progressive policies. They're well-off tech bro liberals that want higher quality of life, lower crime, and lower taxes.