The problem with Greek works is that they were no longer readable (since literacy in Greek disappeared with the fall of the Western Empire), and so there was no interest in copying them.
Not sure where you're getting this from, but it's suspiciously similar to this commentary by Islamic Caliph Omar, where he was talking about the books in the Library of Alexandria.
"Alexandria was captured by the Muslim army of 'Amr ibn al-'As. Several later Arabic sources describe the library's destruction by the order of Caliph Omar.[94][95] Bar-Hebraeus, writing in the thirteenth century, quotes Omar as saying to Yaḥyā al-Naḥwī: "If those books are in agreement with the Quran, we have no need of them; and if these are opposed to the Quran, destroy them."
Not to mention the fact that the Library had ceased to exist by the 4th Century, long before Omar showed up.
(Though I agree with you that the original comment looks like a recycled version of the Omar story.)
1. You're way overestimating the wealth of the Near East in the time period we're talking about. If you look at Maddison's estimates of historical GDP (e.g., https://en.wikipedia.org/wiki/List_of_regions_by_past_GDP_(P...), you can see that circa 1000, "Western Europe" had a GDP of about 10.9 billion (1990) USD, slightly below "West Asia's" 12.4 billion. But by 1500, Western Europe has ballooned 44.2 billion, while West Asia has fallen to 10.5 billion.
2. Italian merchants had regular access to the Near East throughout the Middle Ages and Renaissance, so "suddenly the West had easy access to trade with the East after Lepanto" is not even remotely true.
3. Lepanto's significance even in the context of the 16th Century Mediterranean is highly debatable; arguably, the most it did was to halt further westward expansion by the Ottomans, while they solidified their hold over the Eastern Mediterranean (they did, after all, rebuild their fleet within a year). See, e.g. https://www.historytoday.com/archive/head-head/how-important...
4. The most significant transfers of "knowledge" happened earlier, during the 12th Century Renaissance (translation of Arabic works and Arabic translations of Greek works) and the Italian Renaissance (translation of additional Greek works); continued transfer is evident in the apparent presence of 14th and 15th Century Arabic mathematical and astronomy texts in early 15th Century Italy, where they influenced Copernicus. And, of course, the transmission of other forms of knowledge and technology (compasses, paper, new agricultural products like citrus fruits, rice, and sugar, etc.) occurred throughout the period.
1. The works were still in their original Greek (rather than translations into Latin done long ago, or the translations from Greek to Arabic (and then to Latin) that came in through moorish Spain).
2. Religious attitudes allowed the works to be studied close to their original form. Early Christians were basically of the opinion that if a book disagreed with the bible it was heresy and should be destroyed, and if it agreed with the bible it was unnecessary and could be destroyed. That was less the case in the eastern side of the church and of course a big exaggeration too: there were old Latin translations of eg Plato however they had been quite bastardised to conform with Christian theology (I understand Aristotle was a little less bastardised as early Christian theology was itself quite Aristotelian). By the renaissance less bastardisation was required (if translating to Latin rather than the vernacular).
Without being transferred so late to the west, the works may never have made it to modern times in such a complete form.