If "protecting the public" was truly the goal, why not make taxi driving as simple as getting a commercial endorsement on your license? Driving 50 people in a public bus only requires X hours in a classroom and an extra exam (or two, depending). Interestingly, the medallions were affixed the the vehicle (thus allowing anybody to use it), the total number of medallions were artificially limited and the choice parts of the city were off limits to competition. Does that really protect the public or does it just protect those who own the medallions?
You should look at the history of why taxi regulation came about and why the "right" number of taxi badges per city was a system that much of the world arrived at. It also brought minimum service obligations, driver standards and fair pricing.
Deregulated taxis had cartels, increased and varying prices depending on destination and poorer customer service. Bad enough that many countries and cities found they had to regulate.
Now, whether medallions that are separate from vehicle and driver, and thus have a value in their own right are the best way to regulate is another question entirely. In the US specific case it seems emphatically not. Quite why many US cities had such poor taxi service when other countries have managed much better is beyond me.
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.
