No u.

It’s just not as glorious when your foes have literal sticks!

Was the Byzantine failure to share classical literature down to the great schism, or were there other reasons?

Other reasons. The Byzantines were never particular ‘close’ with the ‘Frankish’ West of Europe, even before the religious divide became serious. The Byzantines regarded the Franks as simple barbarians, and it wasn’t until things became truly desperate against the Turks in the 11th century that they even bothered to swallowed their pride to ask the Franks for help as their coreligionists.

There wasn’t much impetus for the Byzantines to share their libraries with Western Europe. Western Europe was too poor and lacked a developed academic court culture until the ~12th century (Carolingian Renaissance aside), so there wasn’t a lot of demand for books between them and the Byzantines until the Muslim Caliphates had already become the center of learning in the (known to Western Europe) world, and the Byzantine Empire had declined to a dangerous degree.

The Muslim Caliphates, on the other hand, acquired large numbers of Greek texts through trade and conquest (having conquered some of the richest and most developed provinces of the Byzantine Empire), and highly valued textual study, so it was comparatively accessible to them - when Western Europe finally started taking a renewed interest in the classics, it was much easier to find Arabic copies that people were willing to part with than the original Greek texts.