The diet of the ordinary Greek in classical times was largely vegetari

Dear honchos,

My friend, I am happy to respond. Before I respond, I will give you a little more feedback. I do this, not to make you feel bad, but to support your growth and learning. I share feedback with you precisely because I respect you as an intelligent member of this forum and want to support your success in every way.

The first point is a subtle punctuation issue. There is an important difference between the hyphen and the em-dash. The hyphen is a symbol that only appears in the middle of words, gluing two parts together in an hyphenated word:
well-being
able-bodied
broad-minded
mind-boggling
father-in-law
The hyphen is an ordinary keyboard symbol, just to the right of zero on a standard keyboard, and it is a symbol supported by the rtf of this forum. By contrast, the dash or em-dash is a punctuation symbol used to organize the flow of a sentence. In many ways, its use overlaps with the use of the colon. This SC sentence from the OG10 has a em-dash. Unfortunately, the em-dash is not a standard keyboard and there is no way to type a proper em-dash in the rtf of this forum. There is another dash, the en-dash, that is used for dates and ranges of values. They have three different sizes.

The important point is that it is very confusing and misleading to type an em-dash as a hyphen. You typed:
...was largely vegetarian-vegetables, fresh cheese, ...
That makes it look as if you are creating a nonsense hyphenated word, "vegetarian-vegetables." The logic of the sentence requires an em-dash. There is no universally agreed-upon convention for how to type an em-dash in plain text, but I normally use three hyphens (because the em-dash is wider than the en-dash which is wider than the hyphen). This would be:
... was largely vegetarian---vegetables, fresh cheese, ...
I changed the original post to this for clarity. You may choose to follow this convention, but please do not type your em-dashes as hyphens.

The second point concerns discussing individual words. Toward the end of your post, you wrote:
1. What is the importance of as here.
2. what is the importance of with here.
Do you see how confusing this is? When you are addressing individual words, you need put those words in quotes: otherwise, it is terribly confusing for anyone trying to figure out what you are saying. These should be:
1. What is the importance of "as" here?
2. What is the importance of "with" here?
That is considerably clearer. As you may notice below, I use both quote marks and color to talk about an individual word. Please follow this convention, so that you don't create unnecessary difficulty for the person whose help you are soliciting. My friend, I realize that none of this is intended as disrespect. I realize you are learning, and that is perfectly fine. We are all always learning.

Finally, a minor point about my name. My name is Mike McGarry, not MikeGarry, as you have typed. As you go on in life, to business school, in the business world, etc., remember that one of the best ways to show someone respect is to show that you remember their name and that you know their correct name. Getting someone's name correct is always a sign of respect. Again, I am fully aware that you did not intend any disrespect, my friend.

Now, to your questions about this SC problem. This is SC #104 in the OG10. You are correct that (A) & (B) violate parallelism and can be rejected immediately. I think it was very insightful that you looked at the sentences after the list in parallel had been dropped:
(C) The diet of the ordinary Greek in classical times was largely vegetarian, with meat as rare
(D) The diet of the ordinary Greek in classical times was largely vegetarian, meat a rarity
(E) The diet of the ordinary Greek in classical times was largely vegetarian, with meat as a rarity
You are 100% correct: the OA should word as a full sentence with that list omitted.

From here, I will say, (D) is odd, because we just have a comma and then a noun with no connecting word. We can have a noun with no connecting word in an appositive phrase, modifying another noun, but that is not the structure here. Here, we just have an odd noun with no connector, and it is 100% grammatically incorrect. Choice (D) is wrong.

When we insert a preposition such a the word "with," that turns the free-standing noun into a phrase that can grammatically connect to the rest of the sentence. Choices (C) & (E) have this correct.

At this point, it's purely an idiom question. The structure
"with X as rare"
sounds awkward and unnatural. This is idiomatically incorrect. The correct idiom is:
"with X as a rarity."
In general, we use "as" + [adjective] only in a comparative structure, e.g.;
Meat was as rare as mangoes.
They ate little meat, but cheese was not as rare.
There's not really a correct structure for "as" + [adjective] apart from comparisons. By contrast, the structure "as" + [noun] has a few uses, among which is showing that something plays a certain role. The structure
"with meat as rare"
leave us wondering "as rare as what??" It leave us expecting a comparison that never happens. That's the problem with (C). The OA has the structure:
"with meat as a rarity."
In other words, meat is playing the role of a "rarity," or occupying the position of a "rarity." This is idiomatically correct and it reflects the meaning that the sentence is trying to convey. Choice (E) is flawless, a very strong OA.

Does all this make sense?
Mike