To understand the language situation in Korea, let's imagine that we divided the Italian peninsula in half just north of Rome. Does it follow that now there are two dialects, North Italian and South Italian? No. Italian has a bewildering variety of regional dialects, and our arbitrary line doesn't correspond to a genuine dialect border. Similarly, the DMZ cuts across the central dialect region in Korea.
More importantly, Italians will continue to write and be taught in Standard Italian, which was developed based on the Tuscan dialect long before our artificial division of the peninsula. It won't be as if they would start from scratch and create new standard languages based on the Milanese dialect in the North and the Roman dialect in the South. Even independent countries such as Germany, Austria and Switzerland find it useful to use the same standard German as each other, even if it's not necessarily based on a dialect spoken wothin their borders. There will inevitably be differences in vocabulary and spelling, but the differences will be far less than if we imagined a naïve model where each country creates its own standard (which is what basically happened in Scandinavia).
Your comparison of Korean to Italian isn’t helpful, because you’re confusing dialects and languages. Italy has many regional languages including Piedmontese, Lombard, Venetian, Sicilian, Sardinian, Neapolitan, etc., and of course Italian (based on the Tuscan language). Each regional language has many dialects, so for example in Piedmont you have the Turinese dialenct of Piedmontese from Turin, which is distinct from the Piedmontese dialect in other areas. The dialects vary substantially, from one village to another—even different suburbs of a city can have different words for certain things.
Confusingly, Italians would call Piedmontese or Lombard a “dialetto” as much as they would call Turinese a “dialetto”. The word basically means a dialect or regional language, depending on the context. There is also a political element to it—the Italian government has suppressed the regional languages for years, and even now does not recognise them as languages, against academic opinion.
To be clear, Italians would also (generally) refer to Welsh as a “dialetto” of English, despite the fundamental difference of Welsh and English. (In fact they would usually also often refer to the U.K. as “inghilterra”.) The word “dialetto” as currently used in normal Italian speech simply does not correspond 100% with the English word “dialect”, much like the word “camello” doesn’t correspond to “camel”.
The regional languages generally are not mutually intelligible, although this depends on which dialects two speakers speak, and how “stretto” (strong) the dialect is (I don’t know what the academic term for this is). So for example Vercellese (from Vercelli) is linguistically close to Novarese (from Novara) even though Vercellese is classed as Piedmontese and Novarese is classed as Lombard (despite being a Piedmontese city). The distinction is ultimately arbitrary—there is a gradation of dialects from Piedmontese to Lombard. Vercellese for example has many grammatical elements of Lombard (e.g. it uses the Lombard lü (meaning “he” or “him”) instead of the Piedmontese chiela).
Also, an older or more rustic speaker is more likely to speak a “stretto” dialect, because they’ll use more words and expressions originally belonging to that dialect (or to the regional language). Over the years, the regional languages have absorbed many words from Italian, replacing the traditional words. Now, the same thing is happening to Italian with English words (e.g. the word “goal” replacing “rete”, or “babysitter” replacing “tata”, or “shopping” replacing “spesa”—the English word in each case sounds more modern or cool to Italian speakers).
The linguistic situation is basically the same as with Catalan and Spanish. Catalan is as much a “dialect” of Spanish as Piedmontese would be a “dialect” of Italian. In fact, you could just as rightfully say that Italian is a “dialect” of Piedmontese. The difference is political, not academic.
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.
In the case of Italy, I would also doubt (as you do) that a hypothetical “North Italy” would choose anything other than Italian as its official language, because the use of Italian goes back a lot further than the unification of Italy, and it’s the obvious and easy choice as a national language. The government simply doesn’t care about the regional languages dying out, and there’s no reason to think that a new North Italian government would have a different viewpoint. The main motivation, in the case of Italy, was just to have everyone speaking the same language.
However, it’s certainly possible for a country and a population to completely change language in a generation. The Italian generation of people that is now about 60 years old are effectiviely bilingual—they were taught in Italian at school, but (generally) spoke their regional language at home. The generation after that spoke only Italian, and the generation before that spoke only the regional language. (Obviously this is a bit of a generalisation.) So it’s quite straightforward for a country to change it’s language over the course of 20 years or so.
I have a colleague in the Netherlands who believes that the Netherlands will at some point adopt English as its official language. I personally doubt it, but it is nevertheless definitely possible. All it would take is for the government to decree that all schoolchildren be taught in English at school. Dutch people (and particularly Dutch school-teachers, which is the important thing) would easily be able to carry that policy out. The first generation of schoolchildren to pass through to adulthoold would probably choose to speak in English as their primary language, as has happened in Italy.
Also Brussels has changed from a Flemish speaking city to a French speaking one in a short space of time.
More importantly, Italians will continue to write and be taught in Standard Italian, which was developed based on the Tuscan dialect long before our artificial division of the peninsula. It won't be as if they would start from scratch and create new standard languages based on the Milanese dialect in the North and the Roman dialect in the South. Even independent countries such as Germany, Austria and Switzerland find it useful to use the same standard German as each other, even if it's not necessarily based on a dialect spoken wothin their borders. There will inevitably be differences in vocabulary and spelling, but the differences will be far less than if we imagined a naïve model where each country creates its own standard (which is what basically happened in Scandinavia).