I'm not sure I fully comprehend this question: why can there be only one preserver of Greek texts? When knowledge of Greek was lost in the west Greek texts were known to Latin scholars only in translation, and then only what was fortunate enough to survive. Translations of Aristotle were very important, for example, but Plato was largely unknown. This question seems to approach the "preservation" of Greek from a decidedly western point of view, in that it reads to me to understand the society that contributed "most" to the reintroduction of Greek to the west is the only one that "really" preserved knowledge of Greek. Which isn't really how these things work. Both Byzantine Greek scholars and Muslim scholars (mostly in translation into Arabic, but they knew Greek and translated it first) read and studied Greek texts that were unavailable to the Latin west, which didn't even realize that Homer is good because they didn't have Homer, only texts talking about Homer. Both wrote substantial commentaries on Greek texts, many of which were quite important in their own right. Avarroes' work on Aristotle, for example, was crucial in the understanding of the development of Aristotelian thought into late antiquity, which was poorly understood by western and eastern philosophers--Thomas Aquinas held him as the commentator on Aristotle. The so-called Recovery of Aristotle, the transmission of a large number of lost Aristotelian texts from Arabic translations into Latin through Muslim Spain was crucial. Whereas prior to the twelfth century Latin scholars could only read in translation the Categoriae and the de Interpretatione, neither of which is especially great, Muslim scholars helped to introduce some forty Aristotelian texts along with their commentaries to the Latin west, whose scholars avidly consumed these translations--the influence of the rediscovery of Aristotle in large part through Arabic translation cannot be easily understated, given the preeminence of Aristotelianism in medieval philosophical thought. Some centuries later Plato would undergo similar treatment, being introduced by Byzantine scholars and translated by Marsilio Ficino.

But most importantly to view the preservation of Greek texts by Muslim scholars as only existing for the teleological benefit of the west and then weighing that against the teleological benefit of the contribution of Byzantine scholars to the west is...teleology. Muslim scholars did not preserve these texts so that they could send them off to the west, they preserved these texts for their own study, and they had a profound effect on Muslim philosophy, literature, and medicine. That Arabic scholars maintained knowledge of Greek texts--and, moreover, advanced their intellectual worth through larger, and more complete study than was usually possible either in the west or in the Greek east--is undisputed.