Making licenses tradable really doesn't affect the stability of access to licenses or facility to fire drivers in the slightest. You tie the licences to whoever pays for them, and give that entity - firm or individual - grandfather rights to the same number of taxi licences as last year. Sure, if the licences are paid for and renewed by the firms it makes individual drivers dispensable and at risk of unemployment if the taxi firm wants cheaper labour, and if they're paid for and renewed by particular drivers it disincentivises firms from firing then for underperformance; allowing the "owners" of the medallion to collect large amounts of economic rent on a secondary market doesn't change that at all assuming the total number of licenses is still fixed at the same amount.
If anything, arguably it's worse for Joe if Lucky Cabs owns his medallion and also has the opportunity to sell at a profit, and Lucky Cabs are similarly badly off if their drivers own the medallions but don't particularly care for keeping customers happy (perhaps because they're counting down the years til they can sell the medallion on for an order of magnitude more than they originally paid for it).
If you want to encourage flexibility in licenses which are short term leased rather than "owned" you can even give holders the option of pro rata rebates (at the price paid) for trading them in early to encourage people that aren't using their licence to return them.
Basically, the city is in total control of the terms on which medallions can be leased and can collect the full market value of the medallions itself; to not do so is economically indefensible, unless you think giving cash windfalls to its original taxi drivers and speculators is high on the list of civic priorities