I worked for 3 years as a part time taxi driver in first world mega city while doing my post grad at University. Taxis are heavily regulated. If I had refused a fare, I could be taken to court. Especially if the passenger belonged to a certain category like disabled or children needing a ride home after school.
This was pre GPS days so I had to pass 4 different computerised exams about the city and major roads. I had a through police check and had to display my taxi driver license on the dash.
There was proper commercial driver & vehicle insurance that was automatically taken out the day's earning. There was camera in the car including infrared mode. There was a panic button to start broadcasting location, video and audio.
Uber's advantage was completely ignoring any regulations. No insurance. No driver check. No driver safety infrastructure.
And their best move was the marketing bullshit. Even now people like you spout the same talking points.
I put in my card and boom! Done! I see the number on the screen - I pay that number, not that number plus customary tips.(99% of the time completely unearned, BTW)
But of course, the major way Uber "blew taxi companies out of the water" is through their ability to offer below-market-rate prices by losing money on each ride, thanks to unending supply of VC money.
Also keep in mind that the majority of Uber's revenue is from metropolitan areas, not rural areas. And in the metro areas there was a lot of convenience to be gained from their app and matching supply and demand side of the equation and a lot of those areas were underserved by the TLC and specifically car services because taxis didn't even operate there. So they actually took the pent up demand and created a better system.
Now they are still grossly unprofitable, but not for the reason you cited here.
Yes, people forget about the pre-app days when hailing a taxi involved finding a suitable street corner and sticking your hand in the air while often competing with others doing the same. And even in midtown Manhattan, perhaps the best-served area in the country, try hailing a cab during rush hour when it's raining. Taxi services have largely caught up and have apps now, but it was a nightmare for a long time.