Ooh, I like the nice long reply I'm seeing. I hope you don't mind if I tackle it chunk by chunk.
1) You said yourself that the populace is totally ignorant. Your educated opinion is vastly outweighed by countless blind votes who are being manipulated by politicians. Your example does not prove the value of education. Leaders are leaders for a reason: they're intelligent. If you are not in a place of power, your opinion will always be meaningless. If you are really knowledgable about something that could change the world, you will ALWAYS be in the minority so your opinion will never matter unless you are on top.
Your argument here is basically that education doesn't matter; the smart people will always be in charge. I disagree. A stupid voter base is likely to elect either an incompetent fool or a manipulative liar, neither of which generally make for good leaders. An educated populace will be able to evaluate claims made by politicians and vote accordingly. Like you said earlier, individual vote might not make much of a difference, but if society as a whole is knowledgeable, it will be reflected in society's leaders.
And yes, I know what what GMOs are. Scientists are still out on it. The obvious solution is to avoid GMOs until we know more. How much knowledge about biology and chemistry does that solution take?
Thank you, you just proved my point actually. Scientists aren't "still out" on it. The consensus is that GMOs as a whole are fine, and that whether or not they're helpful or harmful comes down to the individual GMO you create. There's no inherent danger in GMOs as a whole, and thus they ought not be avoided. Coming to a conclusion like that requires at least a basic knowledge of GMOs though; what they are, how they work, etc.
2) The phrase "If you do not know history, you are doomed to repeat it" is one of the only reasons history is still taught. Again, in this world, understanding history does not matter unless you are control of a government. You assume that a NONaverage Joe's opinion has any effect. The average wins. The average will always be the target for manipulation. The average will only and always be targeted in its ignorance. There will always be an ignorant area. Also, as a side note, real history is way too complicated for almost any one to understand. Simplicities are ultimately meaningless except for manipulation other people.
This kinda gets back to what I was saying above. You're right: The average wins. That's why I'm arguing for educating the average Joes out there. If you want the right people in charge, you gotta make sure that the majority of citizens are educated enough that they won't fall for manipulation. If properly taught, history teaches the critical thinking needed to dissect rhetoric as well as the basic knowledge to know where we ought to head based off where we've come from. I'd disagree that history's too complicated for anybody to understand. Perhaps no one person can understand the entirety of human history in completely detail, but that doesn't mean that it's impossible to learn, at a fairly complex level, even, the path we've taken to get here.
Basically it boils down to the subjects you picked as important, History, Chemistry, Biology, are not inherently special. EVERY academic area is important or will be important. What everyone should know how to do is think critically. That's why the only three classes that should be mandatory in my opinion are maths, English, and logic. All else comes out of these. Someone does not have to know the intricacies of vaccination to know what side they should be on. People do not have to know quantum theory to avoid an atomic bomb.
We've hit a certain level of agreement here, actually. I agree that critical thinking is, well, critical. However, all logical arguments have to have premises. To come to a conclusion about a topic, say vaccination or GMOs, one needs to have the premises as well as the reason to reach the conclusion, and the classroom's the best place for them to learn those premises. You can say you don't need to know the intricacies of vaccines to know which side to be on, yet I've talked to people who don't know even the very basics of them. Not knowing how they work, they fear them. It's easy to just say "Oh point out the experts, they'll believe them," but the truth is people are inherently afraid of that which they don't understand, and I'd argue that the amount of people in the US who reject topics like biological evolution of climate change is enough to show that the general population won't agree with people just because they're experts.
Smart people will always be there. Unless you do not trust any scientists, you do not have to know anything about history, chemistry, or biology. But! That's only if you want to be part of the system as it is now. If you want to do something utterly revolutionary, you're going to know about these things. But this topic is about things everyone has to know, not the exceptional.
Quoting this here since it ties into what I just said. There IS a certain amount of distrust of scientists in our society. To take an example, although there's scientific consensus on the evolutionary origin of life,
33% of adults in the US claim that life has existed in its present form since the beginning of the world. Since there's this distrust of science, your argument falls apart. At the very least, you need to educate people enough that they'll trust the people who dedicate their whole lives to this stuff, and for that, they need to have at least a basic understanding of these topics.