Uber doesn't stop people having their own clients. Quite the opposite, it's a great place to swap numbers with a driver you like. Uber even has a favourites button so you have a higher chance of getting the same driver/passenger.
Also, your plumber doesn't set his rates, you do. He quotes and you accept (or not). That's exactly the same as the Uber driver.
It seems pretty obvious to me that Uber drivers organise their own work. They start and stop when they want, pick the journeys they want, set their own hours etc. But maybe I've misunderstood you?
The driver does not quote anything, Uber does and both the driver and the customer accept or not.
They have a very limited way to choose the ride they take: they have to make the decision in a couple of seconds, they may not know the destination and they are punished for declining too many rides by Uber. Does not scream free choice of what ride to take.
It is possible to bypass Uber and exchange numbers, yes. But the Uber platform itself does not let you build your own clientele, such as letting you look for the clients you want or the other way around. The question is not whether drivers can build their clientele outside of Uber but inside of Uber.
They do indeed have flexible hours, something that is apparently - reading the comments of this thread - impossible as an employee in the US and thus a feature of being an independent worker, but in France you can be an employee and have flexible hours, so that does not make Uber drivers independent.
Also, your plumber doesn't set his rates, you do. He quotes and you accept (or not). That's exactly the same as the Uber driver.
It seems pretty obvious to me that Uber drivers organise their own work. They start and stop when they want, pick the journeys they want, set their own hours etc. But maybe I've misunderstood you?