A doctor who is able to, would, in an emergency situation.
They won’t in non-emergency situations because of the contractual agreement they and their patients have entered into both with the regulatory agencies and with the insurance companies.
This helps to prevent the “dabbler” or “jack of all trades” trying to do something he can’t do as well as a specialist and causing even worse problems.
There is literally no emergency declaration in your hand being broken in three places, it just means you can’t use it, and that is not life threatening, hence contrary to your point, it will not be set period, until you locate such a doctor and get seen.
Of course the fact that your hand will be crippled for life due to loss of function is not an “emergency” either.
Even the fact that you believe that such inaction is okay-fine shows that the system is not only broken, but is the very epitome of absurdity.