I live in a city, Cleveland, with one of the best doctor/patient ratios in the US. We have two of the top 12 ranked hospitals in the US (according to US News and World Report) and health care is a major industry. I have found your search results to be accurate for here as well, in spite of that.
I have CFS/ME and Fibromyalgia, with CFS/ME being predominant for me. I have seen multiple specialists at both hospital groups here, and multiple doctors in several specialties (Neurology, Immunology, GYN, Rheumatology, Neprhology, Urology, Endocrinology, Genetics, Internal Medicine, Pain Management) at those hospitals as well as other hospital groups here in Cleveland and in other cities (including Mayo) and docs in private practice, and many alternative medicine practitioners in all areas of alt med, all over the country.
What I've found consistently is that all doctors feel that they know something about FM, and most of them know nothing about CFS/ME. In fact, I have found that most of them think FM and CFS/ME are the same thing. Only my immunologist (who is the doc I primarily use, and is in private practice), my internist, and my physical therapist seem to even know that there is a difference. I find that they use the terms interchangeably, and some have said to me, "Aren't they the same thing?". When I speak about my CFS, they respond using the word Fibromyalgia. When I mention exercise and exertion intolerance in CFS, I get a lot of blank looks. I had surgery this summer, and several of my specialists, while I warned them it would take months for me to recover, were shocked that I had not recovered in 6 weeks, and that I am still not recovered to my normal CFS level from it.
I think part of the issue is that there are many more people with Fibromyalgia than CFS, so docs see a lot of it, and are willing to treat it because they see it often. I find that around here, too, there are people being diagnosed with FM that don't have it (my immunologist, internist and physical therapist are all finding that as well, and finding it increasingly frustrating, as they need to rediagnose people with other illnesses, and then get in the middle with patients who are unhappy and confused), so it seems to be a common thing that doctors see, so every doc thinks they know what it is (hence people diagnosing it incorrectly).
I also think that doctors don't want to see CFS patients because there are no drugs that they can give them, so they feel helpless. And a lot of the rheumatologists and neurologists I've seen feel that CFS is not a real diagnosis, since there is no test -- they all believed I was sick, they just weren't willing to accept my diagnosis as CFS, and looked for something else (diagnosed with mitochondrial dysfunction by one, which was good enough for me, since he actually helped me). "CFS is not a real diagnosis," they say.
On finding a CFS specialist, at one of the major hospitals here, their Rheumatology department says that most of their doctors treat CFS, but some expressly say they will not treat CFS patients (those are docs who treated CFS patients for years -- I guess they just got worn out). Even the two doctors I go to who are knowlegable about CFS and accept it and know what it means and is, don't bring me new ideas for treatment -- I bring ideas and research to them (and my opthamologist, whose wife has CFS). Then they share those with their other patients. It's the way of the world with CFS everywhere, I think. I'm blessed to have my immunologist who is there for me, sometimes with weekly calls if I'm going through a crisis, and hour long appointments. I feel blessed for that.