This is factually incorrect. You're drawing a direct parallel between medallion taxis and Uber/Lyft rides, when they are not in fact the same at all.
The outer boroughs, and every neighborhood in the city, has had car service options for decades, where you call a phone number (or often walk to a local storefront stand) to get a ride.
This is the exact market segment that Uber has entered. Literally. There wasn't and still isn't a restriction on the number of TLC car service cars on the streets.
This idea that there "were no taxis" in the boroughs and now there are is ridiculous. There weren't many yellow cabs before and there still aren't. There used to be thousands of TLC cars before and there still are. Except now people use an app instead of calling Arecibo or Carmel.
Street hailing was and still is prohibited, both for Uber and for phone style car services.
The massive innovation, of course, is the use of a smartphone app for hailing, which made it technically possible to create the market that we see today. Uber, Lyft, and the other companies that built internet enabled hailing solutions certainly do deserve credit for building out that innovation, but that change was just an obvious outgrowth of the technological change to everyone having a handheld always connected small computer. It was inevitable.
The dynamic you're describing for NYC just doesn't really fit the facts. Nor does the idea that a lot more people taking a lot more rides in cars is reducing greenhouse emissions, but that's another matter.