I am well aware of the language situation in Italy, and I hesitated a bit before using it as an example. I probably should have qualified my use of the word "dialect" as you rightly point out. I do remind people from time to time that the regional languages in Italy are languages in their own right, and indeed the "dialects" spoken in Italy don't form a natural subgroup of the Romance languages, either (Piedmontese and Lombard are probably closer to the subgroup of the Western Romance languages that includes French rather than the one that contains Italian). But for my point here the distinction between languages and dialects is not important. The point is that in all of Italy, regardless of the local dialect or regional language, people are taught standard Italian. If we divide Italy, the Southern half is not going to switch to, say, Neapolitan, or the Northern half to Lombard, in spite of the rich literary traditions of these regional languages.
The situation in Korean is not qualitatively different, either. For instance, the Jeju "dialect" (now sadly moribund) is definitely not mutually intelligible with other dialects of Korean some authorities would insist on classifying it as a separate language. Even the mainland dialects (as they are traditionally considered as opposed to distinct languages) are considerably different from each other, not just in lexical items but in the existence of different grammatical categories (e.g. distinguishing between yes/no and wh-questions), morphology (the conjugation of verbs and adjectival verbs is all different), and phonology (different consonant and especially vowel inventories, different stress/pitch systems), to the degree that I wonder how much communication would be possible if it had not been for the imposition of standard Korean.
I used Italy as an example instead of the U.S., because it is a stretch to say that the U.S. has different dialects as the speech is quite uniform across the vast country compared to what you see in Korea; or the U.K., where the existence of Scots complicates the analogy. I did not intend to minimize the diversity of regional languages in Italy.
The situation in Korean is not qualitatively different, either. For instance, the Jeju "dialect" (now sadly moribund) is definitely not mutually intelligible with other dialects of Korean some authorities would insist on classifying it as a separate language. Even the mainland dialects (as they are traditionally considered as opposed to distinct languages) are considerably different from each other, not just in lexical items but in the existence of different grammatical categories (e.g. distinguishing between yes/no and wh-questions), morphology (the conjugation of verbs and adjectival verbs is all different), and phonology (different consonant and especially vowel inventories, different stress/pitch systems), to the degree that I wonder how much communication would be possible if it had not been for the imposition of standard Korean.
I used Italy as an example instead of the U.S., because it is a stretch to say that the U.S. has different dialects as the speech is quite uniform across the vast country compared to what you see in Korea; or the U.K., where the existence of Scots complicates the analogy. I did not intend to minimize the diversity of regional languages in Italy.