So Latin spread pretty well across Europe, being the basis of almost every language there, but how come it didn’t take as strong a hold in the Middle East? It’s not as if the Romans were never there, and they lingered there far longer than they did in Europe. Was it just a matter of a stronger preexisting culture, or did the Arabs have something to do with it? What happened here?
Latin would have been used by (most often Latin-native) governors and their administration when reporting to Rome, and there would have been Latin speaking personnel, but they would have been the exception.
So yes, it would have been more accurate to use Greek, but of course it would have made the whole thing a lot less funny because most of the audience wouldn’t have been able to decipher the Greek letters (and it would have added an extra layer of confusion to a clever joke).
Even in the Western half of the empire a lot of the elites spoke Greek. For example: while he would definitely have been capable of speaking Latin, Marcus Aurelius wrote Meditations in Greek.