Any specialty can be taught separately from the start -- we could have a psychiatry school, ortho school, rads school, cards school, etc -- without having to teach all that is taught in medical school. This would produce clinicians that are competent at treating what they have been trained to treat. However, it would cause a fragmentation of the health system. You wouldn't be able to really know and understand issues going on outside your domain. Your ability to understand the research being done would be more limited if it ever discussed the actual biochemistry or effects elsewhere in the body. I think this would cause certain issues to be missed and others to be mismanaged.
Having gone through medical school and taken all those classes has provided me with the background necessary to provide a more 'integrated' care when necessary (and I hate that phrase, I don't mean it in this fluffy, pseudoscientific holistic sense that some do, I'm being more concrete). I certainly don't remember all the details of anatomy or biochemistry, but when necessary I can refresh myself rather quickly as this is information I once knew. And it does matter for understanding some research and some patients. For example, I had an outpatient that claimed to have an autonomic neuropathy due to a cutting of the vagus nerve during a uterine fibroid surgery, and we were wondering if there was some malingering/factitious component to her symptoms. I needed to be able to communicate intelligently with the neurologist, gastroenterologist, and pain doctor. I had to understand what they were saying and the tests they did. I had to use my anatomical knowledge to understand if her claims made sense, and if the Ob/Gyn ever sent me the surgical report, then I would have needed my anatomical knowledge again to make sense of what it said.
For most cases, I believe that most of medical school isn't necessary. But these other cases aren't so rare as to make any of medical school unnecessary for us to learn.