One advantage Uber has over the app you mention is that it's in many (most) cities. You only need /one/ app and one place to go to get fulfillment. The network effect is not insubstantial.
The real advantage Uber has is brand trust when taxis go back to their shady and anti-competitive business practices and the regulatory environment relaxes. The important question is if brand trust will be worth anything.
Sounds like you're suffering from a pretty severe filter bubble.
I'd say it is bad form to base an argument like this as fact, when there is clearly significant evidence of the inverse. We know that Uber has targeted airport usage, with a seemingly offline cache of airport GPS locations baked into the app - to tap the travel market... I'd be hesitant to say this doesn't alone make an impact.
They may not go where you want, they may not be available where you are, because you are not in the right part of the city.
Expect Austin to occur elsewhere (Uber and Lyft leaving due to local regulation, and competitors filling the void).
Uber is not an angel, but at least with one app we have a lot of people pointing out privacy violations. With two (say, Lyft in US cities), there's some choice. If Uber really fails and the taxis end up running the streets again, I don't see any real pressure to make good ride hail / pay apps.