As it isn't a good idea to idealise the Antiquity, so it isn't a good idea to idealise the Byzantines. Yes, we have many texts from Antiquity because of them, but we don't have even more texts from Antiquity also because of them.
I think greek was swirling around still in Europe after the 5th century. Bede must of known some greek for instance. Admittedly in a small circle of People.