No, physics:engineering::logic:philosophy. Or possibly logic:reasoning depending on how you want to do it.
The purpose of an education is to make you smarter than you were when you walked in. Structuring a curriculum around what the not-yet-educated would LIKE to know kind of defeats the purpose—in other words, if your reading preferences were any good to begin with, there'd be little point to taking a literature class.
The first part has nothing to do with the second. Personal taste and understanding have little to do with each other. Structuring a curriculum should be built around what needs to be known to understand what's going to be used. The trivia components aren't necessary.
We might or might not be engaged by what's good, but that's not the latter's fault. There's a perfectly good avenue for someone who'd rather lower his sights in favor of being engaged: it's called a book club. They don't even charge tuition.
Actually, it is the latter's fault. If you can't engage people, how can you tell them much of anything worth while? If you're trying to teach people how to argue, hooking them is part of the components I believe. Failing to hook people is the authors problem, no? Last point: And that's why literature should not be a mandatory component of the general curriculum (unless you think I've been discussing electives or college degrees?).
And finally we move to the blank assertion "popular culture equals culture, and is up to a vote." Congratulations: like many assertions, that's actually impossible to refute. (Of course, like many assertions it's equally impossible to prove.) I'm not really inclined to change the mind of anyone who believes that. Of course, I'm not required to take them seriously, either, and don't.
Actually I can prove it (well, I can state a decent theory, proof is for math): culture (in the sense of works and artwork and whatnot) is a group of memes from random works (language, to some degree, is an extension of this) so what's popular is what's being bounced around by people in general and is what determines general language definitions as language is memetic in nature. Just as one doesn't need to know about War Hammer 40000
to "get" the meme "BLOOD FOR THE BLOOD GOD!", one doesn't need to know the origin of "don't look a gift horse in the mouth" to understand the meaning of the terms. Thus origin of terminology and whatnot is purely trivia as long as the usage of the terms is conveyed consistently.
Fortunately, I'm not required to take any of your views seriously either. Since the OP was about a hypothetical "you" we can each assume our own stances as supreme dictator of education curriculum.
Great authors are people who have something to say and are capable of saying it in the most concise and coherent way possible. You're asking, "Why is formulating your ideas coherently so important?" Two words: common courtesy. Mutual understanding is essential when you're having a discussion, and precise wording helps achieve it — Natural language is ambiguous enough already.
Actually, that's not true. Concise and coherent means not wasting time building a structural narrative, backstory for characters, adding random events and world building in. Essentially, it's essays and links to evidence which is something that's recently picked up a lot of steam thanks to the internet. Having to build a fictional character and trying to invoke emotions to make your point is a flaw in persuasive arguments that only works because the general public hasn't been trained to ignore such arguments. Of which, I am entirely in favor of being the primary goal of writing classes. Since eliminating flaws in thinking is the goal, teaching people to use flaws in the thinking of others is a mistake or deliberately self destructive in the same way a "how to lie to people effectively" class is. It might be beneficial, but it's not really something that should be promoted.
I'd guess about as many people care about pre-calculus as they do about Brit Lit. (It's high school; the social norm is not giving a crap.) Does that mean it, or anything else in the classroom, shouldn't be required, on the simple basis that teenagers wouldn't choose it on their own?
Actually, caring isn't the important aspect or whether or not they'd choose it given the choice. General utility is what's important for picking the mandatory curriculum for general students. The lack of caring indicates a flaw in the teaching style, or much more likely in the curriculum decisions, in a lot of cases (if it's the majority of the students and they haven't been preselected for not caring). Caring is only useful in getting the subject to be retained, so while it's a positive, it's not necessary. As for the specific class pre-cal/trig, yes it should not be on mandatory curriculum of all students. If you're going in for physics in some way, then it's a pre-req, but for any one not going into a field/subject that uses trig, it's wasted time unless it amuses them.