Like I said, Medallions are there to control supply to ensure profitability of taxi drivers. Sounds like bribery? Well not if the government is dictating your price and business model.
The government says "you can't charge 10 bucks a mile when it is raining during rush hour." And note, that is exactly what Uber does when it is raining.
Limiting competition is the compromise for limiting their profitability. The government is ensuring they'll profit, but not too much. It was a common method of regulation in the past.
The problem you're ignoring is that demand vastly outpaces supply which is why the cost of medallions is so inflated. Additional medallions could be sold by the city to increase supply without affecting existing cab utilization or profits but that doesn't happen. Why? Well because the medallion owners rent their medallions out and if the supply of medallions increases then the demand and thus rent paid for medallions go down.
The government says "you can't charge 10 bucks a mile when it is raining during rush hour." And note, that is exactly what Uber does when it is raining.
Limiting competition is the compromise for limiting their profitability. The government is ensuring they'll profit, but not too much. It was a common method of regulation in the past.