> what's to stop entities from outbidding everyone?
They can, but by doing so, they'll pay a lot and enrich the state rather than parasites. On one hand, auctions cost them money; OTOH, they can't charge more for the lease than what a taxi driver can earn. Those are conditions where free market regulation works: if they provide little service, they can only make little profit.
> [some] would get more cheap licenses, and [some] would get exorbitant fees.
As soon as you create a bureaucratic regulation body enforcing byzantine rules, you've created a weak point in terms of corruption resistance. Such an incomprehensible administration is exactly what lobbyists can successfully tune to fit their patrons' interests.
More to the point, the only reason why you might want to regulate the number of taxis is to make sure that every driver can earn decent wages without behaving recklessly. With this respect, the medallion system has been broken, seemingly beyond repair, by parasitic investors. Your proposal doesn't sound like a focused answer to the original problem "how to ensure that taxi drivers earn decent wages without having to behave recklessly?"