Pardon my ignorance, but is there something in the cabbie's contract that restricts her from also being a driver for Uber/Lyft? I assume so; in Japan, this is not the case, because every time I hailed an Uber, at traditional cab driver appeared.
Uber actually has partnerships with some local taxi companies to do exactly that, in cities that treat it as less of a war between the two.
Would you take a Taxi fare for $20 (- $5 medallion rental = $15 profit), or an Uber fare for $15 (- $5 medallion rental you still owe = $10 profit)...
Even if it were legal or within the contract to do so (it may be, I have no idea), it would be a losing prop for a taxi driver to take an uber fare.
Usually, what Taxi's can do that uber can't is pick up riders who are hailing a cab.
That means if I'm walking down the street and see a cab, I can waive it down and it will take me to my destination.
I cannot do this with uber - the only way to get an uber car is through the app.
This is a subtle distinction but it matters and doesn't begin to get into the arrangements cabs have with hotels, airports, etc.
So what OP was saying is that if you're paying for the medallion, to recoup the cost youre incentivized to pick up normal taxi fares and not drive for uber, assuming you can generate the same value.
https://www.uber.com/chicago-drivers/services/uberCAB/
As a customer, I've only done it because I wanted a cab after a concert and didn't want to wait in line -- I walked two blocks and used Uber to hail a cab instead.
You pay meter fare plus tip and the whole thing is charged through the app.