You are citing some problems that aren't "uber vs taxi" but seem to be due to a lack of competition among taxis.
- Taxis shouldn't be allowed to reject you
- Taxis should all take cards, and preferably pre-payment
- Taxis should be bookable with apps
- Taxis should compete for service meaning a nice attitude and a nice car will be common.
So this isn't about "uber vs taxi" this is about monopoly taxi vs. non-monopoly taxi. Once the monopoly is out, Uber will just be like every other taxi company. And better still, all the taxi companies will be like Uber.
It might be that "taxi" in a lot of cities, especially US cities, are monopolies, have shitty cars, no apps, rude drivers etc. But that isn't because they are taxis, it's because they are a monopoly. Uber should be considered a taxi company, because they are taxi. But they should of course be allowed also to operate as a taxi, everywhere they want.