It is far more common for a doctor to underdiagnose a patient that presents with flu systems but actually has the "cool, rare disease" [with the flu than it is to correctly diagnose them the first time].

In other words, within the ecosystem of "cool, rare diseases", it's far more likely to be mis- or under-diagnosed the first time than it is to be correctly diagnosed the first time. One reason for that is they're not common diseases (so the docs won't have a lot of experience diagnosing it), but another fascinating reason is that they have already sort of diagnosed you in their heads before getting into the data.

If you're interested in the topic, you should pick up the book I linked above. There's a fascinating study in there that tests this problem: the study takes two sets of identical x-rays with an obvious primary diagnosis like a cancerous tumor and in one set, they add another problem (blocked artery or torn heart valve or something else obviously serious and life-threatening). They give the x-rays to doctors, inform them of the primary diagnosis, have them look at the x-rays, and track their eye movements. Here's the crazy part: every one of the doctors lingered over that second, undisclosed problem, indicating they saw something wrong. Yet none of them reported the problem. (It's been a while so I might have gotten the specifics wrong, but the gist is there. It's really fascinating and I encourage you to read it on your own.)

On a personal note: take it easy with the "by definition" and "obviously untrue" stuff. I know this is the internet and all but it makes you seem hotheaded, especially when we might be misunderstanding each other.