It appears the basic reason for a taxi medallion system is to regulate the industry and control supply to prevent a flood of random drivers being unleashed on an unsuspecting populous. Uber/Lyft business model: Unleash a flood of random drivers on an unsuspecting populous. Not that I agree with the taxi medallion system as it's ultimately been used to grant political favors and prop up unnatural monopolies but it is…
Not sure if it's a "flood", but the volume of drivers is self-adjusting for the demand. As to the quality of drivers, I have no reason to believe Uber/Lyft drivers are worse than taxis. Chicago taxis drive like lunatics, but this hasn't been so much my experience with Uber/Lyft drivers, who seem to be a cross-section of Chicago drivers. And I don't know that the populace has been meaningfully "unsuspecting".
So all in all, this seems like a net positive--one less thing for the government to spend money regulating in the long run and a better system for everyone else (probably the former caused the latter, but anyway...).