Uber's greatest contribution to the industry is the discovery that customers want a nice and smooth experience. Order using a smart phone, ride in a nice car (or perhaps a luxury one, or a larger one, or one with a child seat - you have options), and don't worry about payment and how much you owe in tolls.
So, my pitch is that this convenience can probably be made to work as a business that doesn't abuse its employees and doesn't flood the roads with idle drivers. And also doesn't burn investor cash in order to conduct a price war. That's just bananas.
I’m sure there are taxi services that offer conveniences similar to Uber, but none have enough name recognition for me to name even a single one off the top of my head. It’s possible that they need marketing help. It’s also possible that they aren’t looking to grow beyond whatever niche they serve now.
This thought experiment is fun, but based on a lot of assumptions like these.