So your argument is that because insisting on doctors getting a license it creates an artificial scarcity. Well in fact it does, for very good reason. Now it just so happens that a bad driver can just as easily kill his passenger as a bad doctor, so there are very good reasons for public safety to insure that certain licensing requirements for operating such a business are in place.
So yes, it does cause a certain measure of scarcity, but there are very good reasons to believe that the trade off is worth the cost. Like virtually, everything in life, there are tradeoffs.
You also point out the weakness of your own argument by saying "we would end up back with some sort of relatively stable model", Precisely, because allowing unlicensed free-lancers in a a business that actually has significant liability and safety costs, that shifts that burden on to everyone else. This is precisely how and why we got regulated taxi services in the first place. Thousands of lives were destroyed before it happened, so necessary regulation its not a trivial matter. Societies that permit some not to play by the rules that others must follow ultimate bear the costs of broken and abused rules. Now you may not see a problem with your wife being raped by a unlicensed driver with criminal intent, or being robbed when all you wanted was to get from point A to point B, or spending the rest of your life in a wheelchair because of an accident the driver had no insurance for, and you are content to just "take your chances". However, the rest of the public doesn't need to be compelled or forced to subsidize your good luck (so far).