Like a situation of economic freedom where millions of individual actors and entrepreneurs can dynamically change the landscape of city transportation instead of imposing a one size fits all scheme?
The goal is to limit the number of cabs on the road without creating unreasonable barriers to other economic activity using the roadways. This can be done without Medallions, though. Cities could just require a license for taxis, like they do for many other businesses. If that license cheap enough that anyone can afford it (perhaps with a small business loan) but expensive enough to slightly deter entry, then it solves the commons problem without leading to cronyism.
A daily "vehicle fee" to have a vehicle within the city might solve your problem.
https://tfl.gov.uk/modes/driving/congestion-charge
Although, as a matter of policy, ultra-low emission vehicles like electric cars didn't have to pay it.
Why? You can't build a close competitor to 9th Avenue (making up a street name) because A) there are buildings in the way beside it and B) building above it is prohibitively expensive. If these are one-way streets as is common in urban areas 9th Avenue's closest competitors are 11th and 7th and after that competition falls off quickly. Five competitors doesn't make a healthy market - it makes an oligopoly.
Your argument is like saying that common grazing land is okay because there's a natural limit to how many cows can be on it; the number of cows that can fit, or that can be watched by the available farmers, or any other limit that doesn't take into account how many cows the field can actually handle.
Medallions don't come close to solving the issue of people free-riding on roads - most of the cars on NYC roads are neither taxis nor ubers.
http://www.citylab.com/commute/2015/05/debunking-the-myth-th...
And that analysis understates the subsidy. In fact it would be far more expensive for private organizations to build those roads due to the need to negotiate with individual landowners. I.e. drivers don't pay for the benefit of the government exercising eminent domain to build roads.
Of course, singling out Uber and taxis for special regulation is a nonsensical solution to this issue; the main beneficiaries are people driving their own car.
Of course, it's important that fees apply to all vehicles. When London implemented a congestion charge, taxis were exempt. It wasn't long before people noticed that minicab registration cost much less than a year of congestion charges. The loophole has been fixed, but it's still an amusing example of incentives.