This might be better answered in a history or linguistic forum; but since I stumbled upon your question while seeking help for myself, I'm giving you an educated opinion, which shouldn't be taken as any more than that.
For me, it seems obvious that this answer to the question, "Why does Ancient Greek have a higher degree of mutual intelligibility with Modern Greek, while Italian has diverged sharply from Latin?" is tied to the history of the Roman Empire (especially for Greek)
The analogy of 'Ancient Greek is to Modern Greek as Classical Latin is to Italian' is, at best, flawed. A better analogy would be: 'Ancient Greek is to Modern Greek as Vulgar Latin is to modern Romance languages'. If you want to be more specific, a better analogy would be: 'Ancient Greek is to Modern Greek as Italian is to the Vulgar Latin spoken in the Apennine Peninsula'.
Many people assume incorrectly--not saying this applies to you--that Italian is the closest living relative of Latin. Phonologically and orthographically, Italian is no closer to Latin than any other Romance language (just ask a Spanish speaker). It's nearly impossible to select any one Romance language as being the "closest to Latin."
As I said above, this question is rooted in the histories of the Eastern and Western halves of the Roman Empire, specifically to the stability of the East and the numerous foreign invasions of the West after the Western half fell (e.g. Gothic influence in Italia). Western Europe was also subject to a series of invasions (and the Plague of Justinian, which decimated the population). Just to name a few examples, there were invasions by Germanic tribes, Huns, Magyars, Mongols, etc.
I'm fluent in Spanish, have studied Latin for six years, and am in the process of learning Italian with the help of my Italian family (Italian as in in Italy). I'm giving you an answer based only on my own perception and my knowledge of Western history; but my perception, like yours, is also colored by my experience.
first of all, is my impression correct? Or it's just a byproduct of me being mother tongue Italian (I can spot more easily the differences and subtleties when comparing against my own language)?
You seem to miss the external factors/pressures that caused Vulgar Latin dialects to diverge into separate languages, largely due to both the isolation of each region from one another and the very large influx of foreign invaders in Western Europe. Don't forget that at the time these languages began to develop, Western Europe entered the Dark Age.
But yes, of course your knowledge of Italian makes it easier to spot differences and patterns in Latin. Native speakers of other Romance languages might disagree with both of us as to which language is closest to Latin; Romanian speakers--although I wouldn't agree--might say that Romanian is the closest living language to Latin, and Wikipedia also makes this connection regarding structure/grammar, for what it's worth.
And since you've mentioned grammatical case, Romanian is the only one of the major living Romance languages to preserve both grammatical case and the neuter gender, which is absent from most other Latin-based languages.
All five of the major Romance languages have undergone major spelling reforms, thus making your question harder to answer. The five most widely-spoken Romance languages all have language academies, so all have had major spelling reforms since the fall of the Western half of Rome.
if so, what historical situations determined this situation? In particular, how did it happen that koiné (200 B.C.) is still mostly understandable by Greeks, while we Italians have serious problems with anything before the "sao cco kelle terre" (963 A.D.)?
My best educated guess would be that the similarities between Ancient Greek, Koiné, and Ancient Greek are due to the fact that the Roman Empire lasted well into the modern era. Remember that Greece was Rome until it was captured by the Ottoman Empire in the 15th century, and that Greeks referred to themselves as Romaioi. While it was subject to foreign invasions, the Eastern Roman Empire was very stable compared to the chaos that ensued in Western Europe after the fall of the Western Roman Empire.
In other words, the Eastern Roman Empire (Greece) preserved the structure/syntax/lexicon of Greek as the lingua franca of the Empire shifted from Latin to Greek. Note that the Renaissance in Europe began close to the fall of Rome, and the revival of Greek and Roman literature was largely due to the preservation of Greek/Roman culture in the Byzantine/Roman Empire. (this is why most medical terminology is derived from Greek). That is, while Western Europe was in chaos and in flux, the Eastern half of the Empire flourished for another thousand years. That's probably the biggest reason for the lack of preservation of Latin.
So in essence, my opinion is that the political stability of what is referred to as the Byzantine Empire today is largely responsible for the preservation of Greek and explains why there is a significantly higher level of mutual intelligibility between Ancient and Modern Greek as opposed to Latin and the Romance languages.
finally, what would be a more correct comparison? Modern Greek is to Ancient Greek what Italian is to ... ? (maybe Dante/Petrarca's Italian?).
As I've stated above, a better analogy would be as Italian is to Italic Vulgar Latin.