This is an interesting discussion. My thought is that there are quite a few variables influencing when literacy/writing becomes an important technology:
a) durability and portability of media: most is either light and degrades easily or durable but heavy and harder to transport.
b) manufacturing cost: papyrus was used for millenia in the west but was never widespread because its production was time and labor intensive. Ink and dye would be another cost. Most stuff just wasn't worth the cost of writing down.
c) population size/density and network effects: small populations wouldn't benefit much because it would be more cost effective to trasmit and preserve information orally. Similarly, large but diffuse populations wouldn't benefit much because the cost of scribes and materials would be high relative to the population served. Given that writing is a "two-sided market" (needing both a trained writer and reader) then the value of writing is somewhat proportional to the size of the network of scribes. Given that scribes are expensive, it would take some time to bootstrap the network. It seems only large and dense populations would see an economy of scale for writing.
It seems like a number of factors have to line up before writing presents a favorable economy of scale for information transfer and preservation.
The Iliad / Odyssey was a purely oral story for a long time. Oral traditions definitely included techniques that would allow any person to memorize vastly more than just a couple sheets worth. For that matter, we still have Hafiz around -- people who memorize the entire Koran and can recite it word for word. Or in non-verbal data, London cabbies with The Knowledge.
Not really quibbling with the point that literacy and writing is important, but the human brain is mutable enough that statements about what the average person "can't" do are suspect.
And it's almost certain the details got changed with each retelling! I wonder if Homer would event recognise much of our modern version of it.
As a former Classicist who studied this in grad school, I can say this is unequivocally false. Rhapsodes of the ancient world had a fairly established canon, by the time of Pisistratus it was in the form we know today. The meter and structure of the poem reinforced preserving its structure and contents so that rhapsodes could recite the poems fairly verbatim. I memorized book 1 of the Odyssey in Homeric Greek as a class assignment, it's actually not too hard, esp. if you do learn with the meter.