By the time students get to Shakespeare and calculus, they've been reading books critically and doing math in school for at least a half dozen years. If they have six years of CS/programming education, too, then I agree that they would be well-served by a course in C.
There are a whole hell of a lot of concepts in computer science that I think are both more useful and far more educational than understanding how to do pointer math and learning how to carefully allocate and free memory for your data.
Actually, they haven't. For the most part, they've never read anything critically (or even really read anything period), and the crank-turning and hoo-ha that goes in math class certainly isn't mathematics.
So by the time they hit Calculus and Shakespeare, they're in year 11 of their time in a velvet jail and have learned to very adequately mimic the bizarre squawking the guards insist is requisite for passage to The Island.
What they have not done is mathematics or reading in any real sense.
I had to read Romeo and Juliet my Freshman year in high school. My understanding is that R&J is a fairly common Freshman text. Your point about Calculus doesn't quite fit either because you don't need an extensive foundation to start programming. That's why we see a lot more 14 year olds programming than doing Calculus. You're a genius if you're doing Calculus at age 14, but I bet a lot of people who read this site started programming that early or earlier. I'm sure we've got our fair share of accelerated math people as well, but they jut are not as common.
My point is that I feel the things that C illustrates and the style of programming it enforces are not that important to learn if you are just starting to program. If you are approaching it from an academic math/CS standpoint, you'd be better off teaching something much simpler like Scheme, and if you're approaching it from a building-cool-stuff sort of hands-on standpoint, I suggest you would be better off teaching Python, Ruby, or one of many other languages with less drudge work (a lot of syntax cruft, manual memory allocation, not very abstract) to do than C.
There may be value to learning C, but I think it's not a good return on investment of your time until you understand many other foundational ideas about programming and CS.
Sure, let's count learning to read, write, and spell correctly.... oh wait, you use that for programming too. ;-)
Also, you should probably revise your thinking so "English" isn't just everything that has to do with books. It's critical theory, complex analysis, and a bunch of other things. The stuff most people do in 1st through 6th grade is to English as learning how to type is to programming.
There are a whole hell of a lot of concepts in computer science that I think are both more useful and far more educational than understanding how to do pointer math and learning how to carefully allocate and free memory for your data.