Actually they were either written by Homer or came to evolve over decades by the telling of dozens even hundreds of people. It does matter, because it’s the proof of how myths perpetuate.
I don’t think ANYTHING can live up to Shakespeare’s hype. It’s an unachievable myth at this point. He’s been lauded to godhead. The basis of English lit should be much wider. They shouldn’t erect pillars, they should put down floors. There were many contemporaries that produced worthy stuff (worthy enough for him to rip off in some cases) plus of course centuries of English literature before a lot of which is really good. English lit is taught in the exact same very bad way American history is taught, much how we jump from Columbus to Plymouth Rock to the Revolution studiously skipping 3 centuries and a few actually very important wars, English lit teacher jump from Homer to Shakespeare and maybe a few mention Chaucer in passing. They really should be finding something to talk about every 200 years or so, there were writers, and some of their writings survive. Shakespeare hasn’t withstood #@$% English teachers’ OBSESSION with Shakespeare survives everything.
The myth of who wrote the Homeric Epics isn’t relevant to how the poems themselves are regarded aesthetically. We don’t know who wrote the original Ballad poems featuring Robin Hood either. But everyone agrees those are of low quality so they aren’t taught.
I’ve spent quite a bit of time with English Dept. academics and, as I’ve said, they go out of their way looking for long forgotten writers to dredge up and be known a the scholar who discovered them. Chaucer is indeed taught to any English major at a decent university. It’s hard to teach him to high school students because Middle English takes a while to master unless you have an exceptional ear for English words...unless you read a translation. All the Arts have ‘Pillars’, Bach and Mozart in music, Michelangelo in visual art, etc.