That's not happening in the next 20 years. My country is still going down the protectionism route when it comes to the taxi industry.
The child seats is the problem though -- as an adult in a city I can easily hail an uber or whatever to get me where I want to go. With kids in tow I'm limited to mass transit, as I can't rely on getting a cab with 2 car seats that will fit. This itself limits the amount of stuff I can take and places I can go.
Perhaps if I had no need for local trips (like yesterday when we went to a canal-side pub 4 miles away for dinner) I could live with simply renting a car for the larger trips -- but to rent a car it involves a walk to the station, an hourly train into the large town, then walk to the car rental place -- all between 8AM and 6PM, then do the reverse on monday morning to drop it off.
Say I want to go to London for a weekend. It's a bit too far for the kids to walk to the local station (especially if it's raining), so we drive to the mainline station, leave the car, and the car seats, in the carpark.
With a taxi I'd have to 1) Hope the seats fit in the taxi 2) Take them on the train (store them where?) 3) lug them on the tube (including a change at Green Park or Bank or somewhere) 4) Go straight to the hotel and check in