But the doctors wouldn't have to compete. If they enter into a coalition where no doctor underbids another doctor, they fulfill both 'serve my own interests' and 'don't be charitable'. Dealing with one doctor becomes dealing with ANY doctor becomes 'every doctor gets everything you have for otherwise fatal or debilitating misfortune'.
There's no 'geographic circumscription on the market' or 'next iteration' consideration in any of Rand's stuff, that I know of, beyond "remake the system in a way that lets us always do what we want and have candy whenever we want", and that's where the philosophy necessarily fails. The second generation of ideologically pure Objectivist society swallows the first alive. Unless they don't breed, since children are little charity-cases anyway, and then the society goes the same way as the Shakers did.