They aren't "classics", they're pre-classical epic poetry. There _was_ a solid lineage of interest in greek and roman philosophy that lasted all throughout the middle ages, and there were latin translations of parts of the Iliad that were taught. The complete Iliad and Odyssey weren't rediscovered from Islamic libraries, but they were carried west by Byzantine scholars. Neo-classicism wasn't a counter culture from Britain -- it was just as popular in London as it was in Washington -- it's not like England was living in the dark ages while America was in the enlightenment.
Homer is a fairly core part of western culture and you see references to it everywhere. Somewhat famously in O Brother Where Art There, but there's a long list of movies and novels that have referenced it. And idioms from it like "a Trojan horse" and "achilles heel" are all over western culture. It's not a biblical-level cultural touch stone, but it's fairly important.
> The complete Iliad and Odyssey weren't rediscovered from Islamic libraries, but they were carried west by Byzantine scholars
To expand, the 8th century Irish were still copying Greek texts and brought them to Germany (Schottenklöster). Southern Italy used Greek through the 12th century, with the Sicilian Norman kings' courts composing a lot of Greek poetry.
Islamic civilizations cared little for the literary heritage of Greece, unfortunately. (Once upon a time, I dug deeply trying to learn how they handled engagement with Paganism. Answer: They didn't, because they didn't read nor translate Homer at al.) They focused on mathematics and science, their achievements came into Europe through Spain (e.g. the Toledo school) around the 12th century, thus Latinized names like Avicenna, Averroes...
Homer is a fairly core part of western culture and you see references to it everywhere. Somewhat famously in O Brother Where Art There, but there's a long list of movies and novels that have referenced it. And idioms from it like "a Trojan horse" and "achilles heel" are all over western culture. It's not a biblical-level cultural touch stone, but it's fairly important.