Non-austenite here. If I arrive in Austin by plane and Uber or Lyft isn't working, knowing nothing about the local alternatives, I guess I have to take a cab.
Have you considered that perhaps, the cost of a taxi fare is close to the cost of operating a car + having a 60 hour-week driver make a living wage (Without VC subsidies)?
I imagine the problem problem you describe will be solved by an aggregator service. You will just open the app and select from amongst the local players.
https://www.google.com/search?q=austin+ride+sharing&oq=austi...
But hey, cabs are fine too. It's your call.