The honest answer — especially for those who have little to no understanding of esoteric grammar rules, but also for those who in fact do — is "there is no reason". It just so happens that English is made that way.
"That not exactly correct" is a perfectly grammatical construction in Russian (or East Slavic languages in general). There, a copula is usually optional, in many cases not an option at all, and in fact for all intents and purposes you could say the word "is" as such does not even exist. (I am exaggerating just a little, and a native speaker or a linguist will understand what I mean. For everyone else, some examples follow shortly, so bear with me.)
Now, there is no intrinsic reason for English to be this way and for Russian to be the other. The only reason is "because reasons". Of course that's a disappointing answer to provide, so you actually can try to drive that point home by pointing out that there are also other differences, in the very same sentence, to which your tutor apparently is not objecting. So it's a small step for him to also not object to this one and just accept it how it is.
For example, note that in Russian (I will just go ahead and use that one as an example), the word "correct" in the sentence at hand would have to agree in gender and number with the subject ("that"), while English does not even have such a thing as inflection of adjectives for gender or number.
- "She is correct, it is correct, we are correct" in Russian would be "she correct[female ending], it correct[neuter ending], we correct[plural ending]".
- "There is a car in the street" would be "On street car".
- "It is cold outside" would be "Outside cold[neuter ending]".
- "Today is my birthday" would be "At mine today birthday".
- "There's something strange about this language" would be "With this language something strange" — where both language and this would be inflected for case, and for a case not just English, but indeed most languages have never heard of: Instrumentalis.
Note how all these examples must have an "is" in English, and must not have an "is" in Russian. From there, it's a tiny step to accept the "that not correct" vs. "that is not correct" discrepancy, as it is indeed the norm and not an exception.
And the reason, again, is pure chance. There is no universal rule that every language must (not) use copulas, or genders, or cases, or aspects, or articles, or the dual number, or a future tense. Some do, others don't. And it took them millennia to develop that way, one tiny step at a time. You can't pin it down to one particular thing.
It's very common in the UK to say 'Let's go for an Indian' meaning let's go to an Indian restaurant and eat Indian food. Historically this is because of the large number of people arriving from India who opened restaurants, a novelty in the 1960s and 70s.
This was brilliantly satirised in the 1990s by the sketch show Goodness Gracious Me where the cast 'go for an English':
One of the more famous sketches featured the cast "going out for an
English" after a few lassis. They mispronounce the waiter's name,
order the blandest thing on the menu (apart from one of them, who opts
for the tastier option of a steak and kidney pie) and ask for
twenty-four plates of chips. The sketch parodies often-drunk English
people "going out for an Indian", ordering chicken phall and too many
papadums. This sketch was voted the 6th Greatest Comedy Sketch on a
Channel 4 list show.
So I guess people could say 'Let's eat American' (singular) - whether they do or not is another matter. America is such a mixture of different cultures that it might be hard to define what an 'American' is, in food terms.
Best Answer
I think lunch does not only refer to the time one eats the food, it can certainly refer to the food. And merriam-webster agrees with me:
So, analogous to meal, breakfast, dinner, on can certainly have it and eat it.
In contrast, this doesn't work with cake:
It seems that with meals, which we can have, the meaning has shifted (or grown) to include also the specific food to be served at the meal, and because of that, we can also eat a meal. But food can be either had or eaten, where having the food does not imply eating it. So the extension of meaning only works in one direction, from meal to food, and not from food to meal. That said, on can order a steak by saying I'll have the steak, please, but strictly speaking, the fact that you probably intend to eat it once they give it to you is only implied.