Linguists concentrate with mental grammar.
Minor gripe here-- the opposition is typically prescriptive vs. descriptive grammar; of course, here grammar is not being talked about as "mental", something that resides in your mind (which is a controversial idea), but as the methodology of describing language structure and use as it actually occurs, without bringing preconceptions and value judgements about how it should be into you analysis.
English teachers may argue that deciding on a set of rules is the only way all English speakers would be able to communicate together but I feel people are smart enough to alter their speech if they want to communicate with a particular audience.
I would argue that, while that may be reasonable for spontaneous speech, you can't really count on it happening at all for writing.
Writing and formal speech are very different activities from casual speech. This is why, IMHO, prescriptive grammar is most needed-- otherwise, people will approach the medium incorrectly.
But a central feature of writing is that the audience is potentially much wider, and that the amount of feedback is minimum; right now, I am writing in my computer, alone in my room, and I can't see your face for your reactions, I can't be interrupted if I write something that won't be understood, etc. Thus, it is much more difficult for me to predict my audience and to know precisely how to adapt my language to it.
Also, there are many differences between the way they speak and that of others which most people just never notice, and pose all sorts of potential problems.
Thus, codifying a standard for language is a sensible thing to do. The problem is that, as you point out, this standard is pretty much always based on the language of a privileged subgroup.
Perhaps it would be less irritable if the set of rules were based on common speech and language practice but that's obviously not the case.
I'd say that a large part of the problem is that there are no institutions with wide membership who get to decide on rules for English, unlike other international languages-- and thus a lot of the prescriptive rules for English tend to be quite irrational. There is nothing illogical about, e.g., negative concord ("double negation"), extraction from prepositional objects ("ending sentence with prepositions"), and a good many things that English prescriptive rules attack. The rationale given for many other prescriptive rules is also typically absolutely absurd; e.g. "to avoid confusion/ambiguity" is typically quite a stupid rationale, since people are exceedingly good at dealing with ambiguity, and typically the condemned usage causes no more confusion than any other.
The Spanish language academies tend to be reasonably good at this-- during the 20th century, Spanish has actually eliminated many orthographic rules. Though a standard Spanish is still much less codified than French or English...
I think I still have a problem. You never say this, but it seems as though you are putting the blame for the problems you mention on the educational institutions of USia. However, while the educational insitution may contribute to this, I think the problem of high and low varieties of speech goes far beyond this. It has to do with the fact that complex societies are divided into socioeconomic and ethnic groups, and that some of those groups constitute the "mainstream" by whose standard everybody else suffers to be judged.