Wouldn't requiring them to carry insurance and maintain their vehicles still be better than government-granted monopolies?

Yes, but it's considerably harder to enforce.

The medallion is what puts teeth on the enforcement of those rules. It's very easy for the owner of a fly-by-night cab company to get new drivers back on the street, perhaps operating under a wholly new LLC with new livery. It's not easy for that owner to get new medallions.

That's just treating the medallion as a deposit. If that's the point then why not use an actual deposit and not artificially limit the number that are issued? Or just treat the cab itself as collateral and seize it for any violation that would have caused forfeiture of a medallion? It may not be worth several hundred thousand dollars, but it's certainly worth enough to have to take it seriously.

see: Fung Wah buses on the East Coast