The main issue is that Austin, like all other municipalities in Texas, is governed by the Legislature of the State of Texas. Home rule cities in Texas have legislatively-granted broad authority but the Texas constitution doesn't protect the actions of home rule cities very much. The legislature of Texas has routinely been very fast to pass laws pre-empting city ordinances they don't like. This is in comparison to the constitutions (and constitutional traditions) of some other states where the legislature is largely expected to let local government do its thing while the legislature concerns itself with the state as a whole.
Or, put another way, Austin might be the deepest blue on the entire continent but that means little when the state government is in direct opposition, and has been for decades.