1) Austin did not ban Uber or Lyft. The city council passed an ordinance that required a gradually increasing percentage of rideshare drivers pass an FBI-approved fingerprint background check. (These rules are in place in other locales in which Uber operates, such as NYC and Houston.) Uber and Lyft formed a PAC that got a proposition on the next local election ballot that would have overturned the ordinance, then spent $9 million on a massive advertising campaign supporting the proposition. The proposition failed overwhelmingly.
2) Uber and Lyft stopped providing services of their own volition within 48 hours. Considering they abide by the same regulations in other cities, it seems pretty obvious to me that it was done in retaliation in order to show other cities considering doing the same that they will actually pull out.
3) Several upstart ridesharing companies immediately jumped into the market and were fully operational in less than a month and are complying fully with the regulations. (Examples are Fasten, Fare, RideAustin, and GetMe.)
4) RideAustin is registered as a non-profit organization, formed by several local tech entrepreneurs affiliated with Capital Factory, an Austin accelerator/VC firm. It is not owned or operated by the city in any fashion.