The answer is primarily based upon imperial geography and its particular connection with the Medieval Greco-Byzantine Empire.
The various Islamic Caliphates, beginning with the Ummayyads in Damascus, Syria, conquered much of the Middle East, the entirety of North Africa, as well as the majority of the Iberian peninsula-(though the Cordoba and Granada Caliphates would eventually establish their own independent Caliphates during the Middle Ages).It is important to mention Medieval Islamic Syria, because of their conquests of Greco-Roman colonies within North Africa, as well as the Middle East.
It remains unclear as to who "lit the spark" and subsequently destroyed the vast majority of the University and Library at Alexandria, Egypt. The Christians have been the historic villain, though there is the possibility that the Muslims may have been the ones who "lit the spark"; it still remains a mystery.
What isn't mysterious though, was the fact that Egypt, like other early Byzantine colonial lands, had major Christian institutions which helped to meticulously preserve the manuscripts from Greco-Roman Antiquity-(most notably, Saint Catherine's Monastery which is located in the Sinai and is supposedly situated next to, "THE Burning Bush"). In Alexandria, Egypt, there is the one of the oldest Eastern Orthodox Patriarchates which exists to this day. Other historic and existing Eastern Orthodox Patriarchates were (and are still), located in Antioch, Jerusalem and of course, Constantinople. Each of these cities-(with the exception of Constantinople), came under Arab Muslim occupation during the Middle Ages. Although the Muslims occupied these particular cities, there were still cultural exchanges between the rising Muslim intellectual communities of the Middle East and the various Greek Christian Patriarchs and Monks-(at Saint Catherine's in the Sinai). The city of Cairo, for example, became a major Medieval Muslim cultural and intellectual center, following the legacy of Alexandria, as well as centuries before the founding of Universities and Libraries in Late Medieval Europe.
The Muslims during the Middle Ages, also had easier access to the famed Silk Route and could quite literally carry with them, loads of precious manuscripts to Baghdad or to Bukhara in Uzbekistan. In other words, Muslim commerce-(whether land or sea based) and the transporting of precious historic manuscripts were essentially, an inseparable reality.
With regard to the cultural blossoming of Islamic Spain, Fes, Morocco, as well as Timbuktu, Mali, the transportation of Greek texts may have traveled via the various North African trade routes dating to the Roman and even the earlier Phoenician imperial age. The sea route is also a possibility with the establishment of major Muslim commercial cities along the Mediterranean coast, such as Tunis, Algiers, Tangier and Palermo-(Yes, Palermo, was conquered by the Arabs during the Early Middle Ages).
It was, however, a different story for Europe during much of the Middle Ages. Remember, the cultural blossoming of Medieval Islamic civilization, was in direct contrast to the European "Dark Ages"-(476 AD/CE-1050 AD/CE). The access to Greek texts within Early Medieval Europe was very limited-(geographically speaking). Except for The Vatican in Rome, as well as Charlemagne's Aachen in Northwest Germany-(circa late 700's AD/CE), as well as the nearby cities of Trier and Cologne, much of continental Europe 1000-1500 years ago, was culturally stagnant and primitive. Illiteracy and religious superstition were widespread, a lack of scientific and medical sophistication were commonplace and the translation of Ancient Greek works were almost exclusively conducted by Monks and Scribes employed by The Vatican. This of course would change with the following events: the Spanish Reconquest beginning in Northwest Spain, the Crusades and (on a less warlike note), cultural exchanges between The Catholic Church and Muslim communities in Toledo and Cordoba, Spain, at the beginning of the 2nd Millennium AD/CE.
Keep in mind, that cities, such as London and Paris, were just mere towns during the Early Medieval period-(when compared with the above mentioned German cities, Rome, the cities of the East, as well as much of Spain). The concentration of power, as well as financial and cultural wealth had shifted away from Rome with the relocation of Roman imperial power to Constantinople by Emperor Constantine-(and to a lesser extent by his predecessor, Diocletian). The redirection of power, wealth and culture would eventually contribute to the decline and deterioration of Rome, but more specifically, to the majority of lands and peoples living to the NORTH of Rome-(with FEW exceptions).
Overall, the survival of Ancient Greek texts is primarily due to the Greeks themselves who created their own legacy of academic scholarship and preservation. However, it was the Muslims, who also preserved the Ancient Greek legacy, but unlike the Greco-Byzantines, furthered the Ancient Greek legacy with great creativity and originality. So before one turns to the Chapter on Late Medieval Europe and "The Age of Scholasticism", please make sure to study the earlier chapters on Byzantium and in particular.......the Medieval Muslim Age.