I'm going to tread carefully for fear of being misconstrued as elitist.
I'm a physician in training. On the one hand, you're absolutely right that patients absolutely need to be their own advocates and should read about things that their physician either prescribes to them or diagnoses them with. An educated patient is a safer patient as they can minimize the occurrence of these errors or, if they do occur, make a fuss to correct them. I'm thrilled when patients ask great questions and are eager to learn more about what's going on with them. Nothing makes me happier than doing some teaching with patients when they're genuinely interested and invested in their own care.
On the other hand, the layman really doesn't have enough medical knowledge to suggest treatment plans or otherwise intelligently question why management plans are what they are. This is because the patient - and the lay public generally - have absolutely no idea what all is involved when it comes to just "giving me an antibiotic" or "giving me a CT scan" or any other myriad of interventions that people think they need. The lay public thinks medical diagnosis is as simple as typing in their symptoms in Google, clicking on the first WebMD article that comes up, and voila! I have a diagnosis! This is so absurd and yet seems to be what the average person thinks medical diagnosis is all about. I would remind you that the Hubski userbase is hardly representative of the general public at large.
Just the other day I had a patient who refused to have an IV of appropriate size placed in his left arm prior to a surgical procedure because "that's not what they did to me last time." We made it clear to him that IV placement varies depending upon what's accessible at the time, and despite warning him that his refusal could very well result in his death due to an inability to push fluid quickly enough in an emergency situation, he was unrelenting. What are you supposed to do there? This is the education level with respect to medicine that physicians deal with on an average basis.
There's a reason physicians go to medical school for 4 years and complete a residency of a minimum of 3 years before being allowed to practice on their own, at least in the US. Please note that that's a minimum of 7 years of additional training beyond post-secondary education. Trying to explain the nuances of management and diagnosis is difficult because the lay public doesn't have the background necessary to explain these nuances to a degree that they will be satisfied with. Let's take a look at the article you linked to and discuss why some of these diagnoses may be missed.
Pneumonia: typically presents with fever, shortness of breath, and a productive cough. Other things that present this way, no name a few: viral upper respiratory infection, bronchitis, bronchiolitis, fungal infection, lung cancer, and many others. All of those things are dependent upon the history that particular patient gives you, and each requires a different management plan. Ideally the physician would've ordered a chest x-ray which is a dead giveaway for pneumonia, but we can't x-ray everyone who presents that way. So that leaves us with one of two options: treat conservatively, i.e., assume it's a likely viral infection (more common and resolves with no real intervention) and have the patient return for follow-up in a few days, or treat empirically with antibiotics and the risk that entails (inappropriate antibiotic selection, antibiotic side effects, etc. etc.).
Decompensated heart failure: this is actually pretty surprising as this should be a slam-dunk diagnosis. The typical patient is obese, has a history of chest pain/shortness of breath with exertion, and is older (40s-50s). They might also have some swelling in their legs (peripheral edema) though not always. Again, you could get a chest x-ray which would show some fluid in the lungs (indicative of the heart not working well), but depending upon how the patient presented you may not do that. What if the patient just tells you they feel tired and short of breath? That could just as easily be pneumonia or some kind of respiratory diagnosis, though, again, I'm finding it difficult to understand how CHF could be entirely missed.
Acute renal failure: this typically presents with oliguria or anuria (i.e., little or no urine production), though clinically this is difficult to diagnose. Diagnosis requires labs. Assuming labs were sought, this should once again be a slam-dunk diagnosis as it'll be pretty evident, but without labs it'd be difficult. You might think the oliguria/anuria could be due to some kind of obstruction along the urinary tract, though without any history of pain or blood from urinary tract that would be highly unlikely. Theoretically a physician would order basic labs as well as a urine analysis if a patient told them that they're oliguric/anuric, but again, the devil is in the details and in how the patient is presenting. Depending on what the patient is complaining of or telling me, that may not be the first thing I jump to.
Cancer: this is a pretty damn difficult diagnosis to make in the primary care setting as an initial complaint. The possible presenting symptoms are immense, and the only real clues you might have are vague complaints of fatigue and unexpected weightloss. If the cancer is metastatic, they might present with bony pain (metastases to bone), headache (metastases to the brain), or some kind of organ dysfunction (e.g., liver mets might cause right upper quadrant abdominal pain, lung mets might cause shortness of breath or hemoptysis [coughing up blood], etc.). In the absence of knowing the specific cases and how they presented, missing a cancer diagnosis would not be unexpected. Once again, depending on the patient's history it may not be the first, second, or third diagnosis on your mind. What if the patient was an otherwise healthy young man or woman? What if they're diagnosed with a lung cancer despite no smoking history? Cancer is the great imitator that requires what we call a "high index of suspicion" in order to diagnose in some patients. I'm not going to fault the 6% of primary care physicians that failed to diagnose it.
UTI/pyelonephritis (infection of the kidney): UTIs are theoretically easy enough to diagnose, and often if suspected patients will be sent home with an antibiotic and no further testing. Pyelonephritis, though, is not necessarily so obvious, and if you suspect a UTI the antibiotics used to treat it will likely be insufficient to completely treat a pyelo. Because they're getting slightly better though - the antibiotics will still do SOME work - the patient might think they're getting better and not think to let the physician know that they haven't improved a week later.
I give you all of these details simply to give you a very limited snapshot at how complex diagnosing supposedly common disease is. Patients don't show up with signs telling us their symptoms, and the same disease will present in a different patient in a different way. Telling physicians to "simply diagnose better" is like telling pilots to "simply not crash the plane" or mechanics to "simply fix the car." To those not involved in those fields the solution is obvious, but, as with many things, the problems lie in the particulars.
Theoretically these things are all controlled for in this study design, meaning that these errors which could've been due to diagnostic difficulty are actually bona fide errors. I have no way to say one way or the other and have to take the authors at their word since we can't see the data they used to design the study. But I think it's important to keep these things in mind when critiquing physicians or demanding better accuracy. Physicians are by no means immune from criticism, but I also think it's important to keep in mind that the lay public is, as a general rule, not sufficiently educated or trained enough to understand many of the difficulties involved in being a healthcare practitioner. This is how, for example, nurse practitioners have been granted the ability to practice independently after 3 years of formal training (with a total of 600 hours of clinical experience) while vehemently denying the need for additional required clinical training similar to that of physicians, calling it unnecessary, a waste of taxpayer dollars, and limiting their ability to alleviate the need for healthcare practitioners (read it for yourself here: http://www.aanp.org/images/documents/policy-toolbox/nproundtablestatementmay6th.pdf). Any physician will tell you that's absolutely absurd, because despite our 4 years of training with 1000-2000 hours of clinical training, no medical school graduate would feel comfortable treating patients on their own. But the average lay person doesn't get this, and because they haven't had the experience themselves it's difficult to make the arguments in support of things like this.
I know I went on a bit of a tangent there, but all that's to say: trust your doctor, not Wikipedia. Be an advocate for yourself, but also recognize that the physician treating you has had a minimum of 7 years of training in medicine in addition to the years of experience in practice assisting him in the process.