Full disclosure: I am a practicing physician.
First off, can you clarify what you are intending "de-professionalization" of medicine to entail? Are you looking more "non-traditional" physicians? Or are you looking to change the underlying structure of the profession of medicine, i.e. the fiduciary relationship between and physician and their patient? That is such a vague term I'm not sure what you're actually asking.
In this ATL, about two-thirds of U.S. doctors have worked as nurses. Of course, they have. How else would you learn medical practice? Just like most auto sales managers have worked as sales people.
Couple of intrinsic faults to this logic.
First, being a physician is not just being a manager of a group of nurses (there are actually nurse managers whose job that is). Nursing and medicine (defined here as physicians) are fundamentally different professions. The overall goal (patient care) is the same, but the approach and mindset are distinctly different. In western medicine these approaches are synergistic and overall care would be markedly poorer if only one approach was dominant.
Second, physicians are not the only "professionals" in the medical field. Nurses, therapists, audiologists, psychologists, techs etc all have professional organizations and significant educational barriers to entry. So this is really just one professional becoming a different one.
Lastly, if you're looking for "non-traditional" physicians, you don't have to look very far. A small but significant percentage of doctors come from many previous careers from both inside and outside the allied health professions. I have colleagues whose previous careers include a paramedic, economist, musician, and combat engineer.
As I understand it, a lot of medicine is trial and error in a respectful sense. For example, if someone has a tricky kind of pneumonia, hit it with one narrow-spectrum antibiotic, and if that doesn't work, hit it with another. You might be running a culture at the same time, but in the several days it takes to get back an answer, you can try your best guess antibiotic. And whether a particular antibiotic works or not, is itself diagnostic.
I saw a doctor with a cough, sore throat, and maybe a chest cold. I came back to see him about a week later still having the cough. The guy was a little put out to see me. Well, these are the cases you want to embrace. The hard cases are the good cases.
In general, that's an outdated view of medicine. As recently as 20 years ago, the majority of treatment decisions were based on physician experience and anecdote. So yes, educated guesses and a form of tailored trial and error. However, there has been an explosion of data on evidence-based best practices since then which has transformed medicine. For the preponderance of diseases that affect most people, for example diabetes, high blood pressure, high cholesterol, we know the ideal treatment (at least at the population level).
True trial and error, empiric medicine is reserved for the so-called "zebras," rare diseases where we either don't understand the pathophysiology or have an available treatment. These are the diseases you see on all the medical shows, which skews perceptions of how physicians practice. In the real world, the few physicians who practice like a Dr. House charge cash up front and sure as hell won't take people with Medicaid/Medicare. Your average primary doc needs to see 25-30 patients a day just to stay afloat financially (the Medicaid reimbursement from a standard clinic visit is about $25-50 depending on time/complexity).
In regards to your experience, it sounds to me like you just had to deal with a physician who was overworked, uninterested or both. If he/she truly just threw a random antibiotic at you for a week of cough and sore throat then honestly, they're not doing a very good job. Most of those symptoms are viral in nature, so an antibiotic is ineffective. Even if it is a bacterial pneumonia there are specific antibiotics that are first-line, and we usually start broad and narrow our antibiotics as we get culture and sensitivity data (other way around is a prime driver of antibiotic resistance). Obviously I don't know the details, but it sounds to me like you had the misfortune to deal with a doc with poor bedside manners and mediocre clinical acumen.
I can damn sure say there are plenty of bad or middling doctors in this profession!
Again, trial and error in a respectful sense. Which is kind of the opposite of the school skill where you really focus on "being right." or maybe school serves as a proxy for a generalized IQ test, even though we now know that intelligence is multiple dimensions and these tests aren't near as good as we once thought.
And the defining characteristic of a profession is supposed to be the self-policing aspect. Which occasionally does happen in the punitive sense of suspending a doctor's license, but not in the more positive sense of coaching people up. Not like it should be. The norm is that doctors go it alone. You can't criticize another doctor even if you're 90% sure he or she is doing something seriously sub-par. It's like the profession lacks the medium social skills and the medium courses of action.
Agree that school performance is an unreliable indicator of both intelligence and success as a physician. Problem is it's hard to test for the social skills and emotional intelligence necessary to make a truly excellent doctor. Also, plenty of medical students start out bright and altruistic but sucumb to burnout and dissatisfaction.
Self-policing is a bit of a mixed bag. There is some, but there is unspoken etiquette in the field that puts up significant barriers to physician-physician criticism. It's certainly an area for improvement. However, you're quite off the mark about the lack of coaching. Almost all physicians undergo a prolonged apprenticeship (residency and fellowship) where this exact process occurs - it ranges anywhere from 3 to 9 years depending on the specialty of the physician.
Finally, I'd argue that the true marker of a profession is the fiduciary relationship, i.e. the responsibility of the physician to subordinate his or her interests to that of the patient. At least, that's what attracted me to it!
Anyways, kind of a massive post. Would love to hear specifically what aspects of medicine you're looking to change - there's plenty of room for improvement.