(Native American English speaker here.)
I think you are right that “the” could rightly appear before “dishes”. However, I find the sentence slightly better without an article there. I’ll try to explain why. Hopefully this will shed some light on how articles work in English, especially why rules can’t always explain it.
Why “the dishes” makes sense
First, here is why “the” makes sense before “dishes”. “The” refers to a specific instance of the following noun, especially a specific instance that was recently introduced into the discourse (or instances with a plural). “Dishes” have not previously been explicitly brought into discourse, but “Korean restaurants” have. A reader reasonably understands “the dishes” to refer to “the dishes served in Korean restaurants”.
Here are some similar examples of “the” referring back to an instance that was not explicitly mentioned previously, but referring to something closely related to what was mentioned previously;
I took my car to a garage for repairs today. The problem is the water pump.
Because you mentioned that your car is getting repaired, “the problem” refers to the specific problem that is the reason why you took your car to the garage. You could also use “the” to refer to the address of the garage, the name of the garage, the model of the car, and probably many other things that a listener might reasonably think are relevant—there can be no exact rule or boundary. The only rule is that what you refer to with “the” must be specific to what you introduced into the discourse. If you used “the” to refer to a problem that had nothing to do with your trip to the garage, that would be wrong—your listener would misunderstand.
You just heard Tchaikovsky's Violin Concerto, Op. 35. The soloist was Julia Fischer.
I’ve tried to make kimchi five times now, but I still haven’t gotten the spices right.
Why “dishes” is slightly better
The example sentence is about “Korean restaurants”. There’s no article, so it’s talking about this type of restaurant in general, not specific Korean restaurants. So, when you reach the word “dishes”, you are still in the world of abstract types, not specific instances. To indicate that you want a plural noun to refer to the abstract type, not to specific instances, you use no article. Notice that you could also say it this way:
The vast majority of dishes in Korean restaurants involve rice, soup and an accompaniment of side dishes.
In the original sentence and in this one, “dishes” is regarded abstractly. (Even in this sentence, you could say “the dishes in Korean restaurants”.)
You might reasonably object that the original sentence is not really about Korean restaurants in the abstract, it's about a specific set of Korean restaurants: only those in Korea. So, shouldn’t we think of the dishes as the specific dishes served in those specific restaurants?
Indeed, that is a legitimate way to think about it. However, the way the original sentence is worded, with no article before “Korean restaurants”, it asks you to think about them abstractly. So, it's a little more natural and harmonious to continue the abstract wording (no article) when you get to “dishes”.
This is not a rule
In this sentence:
The Korean restaurants of Seoul are still the most popular in the city. The vast majority of the dishes involve rice, soup and an accompaniment of side dishes.
“the dishes” is probably a little better than “dishes” because you are referring to “The” Korean restaurants of Seoul. If you said “dishes” with no article, a listener would momentarily wonder if you have changed the topic to dishes in general. But it's still okay to omit the article. A listener will understand that you are referring abstractly to the kinds of dishes served in the Korean restaurants of Seoul. Omitting the article here is still normal English.
Both of these are okay, though I think the second is slightly clearer:
Kimchi can be tricky to make. Spices can be hard to find.
Kimchi can be tricky to make. The spices can be hard to find.
In other situations, though, different customs can apply. For example, this is wrong:
I’ve tried to make kimchi five times now, but I still haven’t gotten spices right.
That would sound like you have switched to talking about spices in general, not just spices in kimchi. You can’t even omit the article when explicitly talking about kimchi in the abstract:
When making kimchi, it can be difficult to get spices right.
When making kimchi, it can be difficult to get the spices right.
but:
To make kimchi, you need spices that you can’t find in this supermarket.
To make kimchi, you need the spices that you can’t find in this supermarket.
In these sentences, other factors override what happened with “dishes”. In the phrase “get X right”, omitting the article would lead to X being understood generally, not in relation to something previously stated. With the verb “need”, “the spices” would have to refer to specific spices already mentioned, not merely associated implicitly with kimchi.
So, there’s no general rule. This can be frustrating if you’re trying to learn how to use articles! The way to learn them is to (1) learn the most common and representative usages; (2) become sensitive to the factors that influence how people interpret and expect them in different situations; and (3) become accustomed to familiar phrases, which influence those expectations.
For example, the common phrases “get the details right”, “get the facts right”, “get the answers right”, etc. influence people to strongly expect “the X” in that context to mean “the X that pertains to something else you’re talking about”, so if “the” is omitted, it suggests that the speaker means X very generally, not in relation to something specific. I recommend not memorizing very many of these. As you get accustomed to them through actual use, you gradually learn to make analogies with them in new situations the same way native speakers do.
