I'd say it's more a matter of setting standards so that drift is reduced and ESL learners don't go bonkers with a dialect that doesn't generalize well, than as a mark of education. For example the differences are already bad enough between American and British English.
ESL here, difference between American and British is nonexistent to me. Sure, some of the word choices differ slightly, like torch vs. flashlight, or rucksack vs. backpack, and even then most of the time, both languages have these words in dictionary, it's just the default that's different. And ESL people are usually in the exact situation you describe, because they're being taught British English during lessons, while learning American English from everywhere else - TV, videogames, Internet. We manage just fine.
Still, given that everyone gets taught grammar for their native language at some point, too, I agree this is in part to reduce language drift. Makes sense - a modern nation with millions or tens of millions of citizens need institutional means to maintain social coherence. Formal education here (at primary/secondary school level) is less about marking someone as educated, and more about ensuring that people from opposite ends of the country can communicate just fine, because without it, their dialects would drift apart within few generations.
