1 - People relate fat the nutrient to fat the body type because they share the word "fat". Not much to do there. In high school I was taught that carbohydrates were "energy" and fat was "waste". People should probably be taught the right way our biological systems operate.
2 - Medical authorities have worked with the best science they had available at the time, and are almost always managed by bureaucrats, not physicians. This isn't limited to the governmental level; hospitals, medical organizations, medical schools... the people in charge often have administrative training, not medical training. As a subpoint; sure, medical authorities have led us astray with nutrition information, but let's not jump to the conclusion that they have done something intentionally evil here.
3 - It's hard to prove any one way is the right way because of so many variables, including genetics, activity level, co-morbidities, and so on. More importantly, when something is right for 95% of people, that also means it is wrong for 5% of people. In the absence of easily testable datum, those anecdotes get a lot of attention. And even if it were easily testable, 5% of the time it would be wrong.
To me, the key is two-fold:
1 - People need to be educated about the facts of how their biological systems function. They need to know how calories work, insulin and blood sugar regulate, and gain a general appreciation of the science of their bodies.
2 - People need to be willing to "listen to their bodies" and experiment. That's the only way to know what really works for you, because you ARE different than everyone else. A healthy diet isn't just about finding mathematically optimal foods, it's about finding stuff you like to eat, that fills you up, that you can eat in moderation, that is easy to make. You need to be willing to mix health and convenience, taste and availability, etc... and that takes a bit of time to figure out.