Some have already done it, Taxi Stockholm has a very nice app; enter pickup and destination, book it, track it, pay through it, etc. And they're marketing themselves by stressing all the things they provide that Uber doesn't. You get a licensed, properly insured taxi driver in a nice car.
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.
Most people use Uber almost entirely in a single city. This is obviously not true for those who travel a lot, but for most people they only need a single app (whether or not it be Uber) for their home city.
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.
But does Uber really have a competitive advantage here? In the multiple cities I have lived in in the past couple years (Boston, Austin, Dallas) I have never noticed Uber's range being greater than any particular taxi. I do prefer uber do to brand trust, but I had no issue switching to RideAustin after uber left the city.
So the people I know that use Uber in multiple cities (mostly for work), spanning multiple employers and multiple countries and locations is a 'filter bubble' - but the inverse perception from you is common sense?
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.
Spend time: figure out what app, download it (possibly on data roaming), make a new account, use app once, be stuck wuth data breaches forever, and good luck figuring out which one is tracking you all the time and running your battery to the ground.
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.
I don't travel frequently. When I do, install the app at home and then uninstall on return. So they can't track me when I am not using them. I understand it's not for everybody.
Google for it and then stand around reading the reviews to figure out which is the best one, you mean. And sign up for a new account and everything with the new app. By that time I'm already on my way to my hotel with Uber.