Uber and friends have indeed democratised giving rides to people - though where I am, a few rich people have bought numerous cars and have daily wagers driving them finding fares via Uber - but at the cost of far more cabs on the road.
Others, notably motorbikes and scooter ride aggregators have emerged to replicate Uber. These motorbike cabs are even harder to regulate than cabs.
Uber, imo, has broken the equilibrium that existed before.
Uber has absolutely increased traffic levels in the places where they operate. I don't personally think it's to a level that is actually a problem, but I also avoid driving myself around in cities whenever I can, so I may not be the best at observing this.
You used driving as an example. I don't see how it's better for the city if we encourage more people to own and park cars because taking taxis is too hard.
The nice thing about Uber is that it lets those cars used for personal driving double as taxis when demand is high. Instead of having your own car and parking it, you can have someone else use their car to drive you.
I think some people have become so desperate to make Uber evil that we're intentionally ignoring what's gained.
Finally, regulating taxi medallions is an awful way to address traffic. If traffic is the problem, you regulate vehicles and traffic. Creating a system where you have to buy something approaching the price of a house just to do a job with a low hourly rate because the city wants to regulate traffic is beyond broken.