To get a better understanding of how podiatry and dentistry came to be you ought to look into the history of medicine. It is certainly more complex than what I might post here. I really recommend you do some reading on your own.
Way back when, the "Medical Doctors" had a more limited scope than they take today. They considered themselves to be "educated" and many things were below what they would do. Medicine was cerebral rather than a "hands-on" type of thing. That would have been menial, and not for persons of their standing. They felt that surgery and such was beneath them. For a reference to this, I would point out that the Hippocratic oath states they will not take up the knife to cure kidney stones. That this is better left to those who do that sort of thing. Think about the "Barber-Surgeons". Treating ailments of the foot, (Corns, calluses, etc.); or pulling teeth was beneath them. People still needed these things done, so people other than the "Medical Doctors" did them. For many obvious reasons, what they could do was limited. For example, without anesthesia there is only so much that can be done.
To this day, there is still some rivalry between "Medical" Doctors and Surgeons. If you spend some time among both, you will notice some differences in temperament and personalities. There are jokes about it, and there is some truth to it. The Dentists and Podiatrists both fall outside of this as they are newer "specialties."
Things change with time. Podiatry came out of the roots of Chiropody in the past 100-200 years. With the advances made in antibiotics, anesthesia, surgical technique and so forth, Podiatry has become a more surgically oriented specialty than could have been envisioned by the chiropodists of 1900.
In addition to this, some podiatrists spent a great deal of effort putting together their theories on biomechanics. This is more recent, like early 1950's. Lots of stuff going on with biomechanics. This is one major focus of podiatry that sets it apart from the MD/DOs.
I think it is an interesting topic. Unfortunately there are few sources of podiatric historical literature. On the other hand, it is a young profession, so journals are available as far back as the profession began in this country. The big questions concern what happened before then.
Hope this helps a bit.