None of these things are reasons for a limited number of medallions.
To be clear: a medallion system says, "In San Francisco, we will have 680 cabs, no more."
That is separate from the question of whether you license cab drivers or charge for the license.
You could say, "In order to drive a cab in this city, you must pass an area knowledge test, pass a more-onerous driving test, you must re-pass these tests every two years, and we will revoke your license if you're involved in an accident. And in order to defray the costs of administering this program, we will charge you $1,000 for that license. And you must use this pricing scheme for fares." But if you were willing to license every person who passes those tests and pays that fee, without regard for how many people have already passed the tests and paid the fee, you aren't operating a traditional medallion system.
To be clear: a medallion system says, "In San Francisco, we will have 680 cabs, no more."
That is separate from the question of whether you license cab drivers or charge for the license.
You could say, "In order to drive a cab in this city, you must pass an area knowledge test, pass a more-onerous driving test, you must re-pass these tests every two years, and we will revoke your license if you're involved in an accident. And in order to defray the costs of administering this program, we will charge you $1,000 for that license. And you must use this pricing scheme for fares." But if you were willing to license every person who passes those tests and pays that fee, without regard for how many people have already passed the tests and paid the fee, you aren't operating a traditional medallion system.