Good doctors encourage patients to understand their condition. Let's not conflate that with exasperation over people who have a week old cough that they've decided is lung cancer because they've spent too much time on wrongdiagnosis.com.
No matter who you hire, in any context, it is up to you to make sure they do a good job with the tools available to you.
You are completely right about the "wrongdiagnosis.com" conclusion. Be aware of your health, and the general statistical chance of various diseases (heart/diabetes/alzheimers etc.). Also get a second or third opinion from other reputable doctors if you disagree. The chances of them all being wrong is very small.
Those examples you hear of people going to "10 doctors who all say they're fine when they in fact have a rare disease" are, by definition, insanely rare.
It's usually the flu :D.