What you described is basically how it works. Getting a license to drive a taxi requires just a commerical endorsement: https://dmv.ny.gov/driver-license/get-license-drive-taxi-or-.... It's the vehicle itself that requires the medallion.

If you want to argue with the phrase "protect the public," that's fine. The point is that, pre-smartphones, the medallion system ensures a functioning system, a modicum of safety, no fare surprises. It was not intended to produce the _lowest possible fares_ for riders.