You can't immediately tell from the title, but then titles are not typically used by an individual to broadcast their occupation - we don't have variants of "Mr" for plumbers, bank managers, or rock stars - despite their very different occupations. Rather, the title is to be used by others when addressing that individual, in order to signify a degree of respect, typically for a particular level of training, qualification and responsibility, or else for a particularly respected position in society. Even the term "mister" is a meaningful sign of respect that historically would not have been as widely applied as it is today - the ratchet of etiquette has gradually eliminated everything below it.
The actual title "Doctor" means "teacher" (from Latin "doceo", "I teach"). This title is more often more relevant to PhDs than MDs, so you probably have your suggested solution backwards. That said, the solution is really neither necessary nor appropriate. Much like "Master" (from Latin "magister", in this case "teacher"), "Doctor" signifies that an individual has not only gained enough competency to practice in a particular field, but has developed enough expertise to instruct others. An individual who is sufficiently qualified to practice but not teach would historically have been known as a "journeyman", roughly equivalent to "professional".
In short "doctor" refers not to a field of expertise, but rather to a level of expertise.
Incidentally, most UK surgeons drop their title of "Dr" and revert to "Mr" after joining the Royal College of Surgeons. I've heard through a friend of at least one surgeon who reacted quite angrily at being addressed as a mere "Dr", which in such circles, due to a collision between traditional titles and modern medical training, could be unkindly translated as "trainee".