Kapten and Bolt already exist, and give you access to substantially the same pool of drivers. Maybe some drivers are on Uber but not one of the others; that will change very quickly if Uber are banned. There's also Addison Lee, who have their own drivers, but offer a similar experience.

This tunnel vision of only seeing one provider of a new but fundamentally commodity service is interesting. It reminds me a lot of how Git and MySQL took off - people encountered them, thought, "oh, that's great, i'll use that", and never stopped to think if there might be better alternatives.