Scholars should be involved in choosing the great books just as doctors choose the best medicine. Parents can effectively guide education up until the teenage years but after that one needs an amount of expertise in a wide variety of subjects that most people can’t provide individually. A lot of very educated people had illiterate parents.
Your problem there is that picking “great” books is opinion based while picking the best medicine is adult based. And as we’ve already discussed to death the book “scholars” are picking as great tend to be books only they want to read and normal people only read when forced to in school. The “great” books wind up being one of the primary reasons people avoid school and hate reading, largely because scholars like them for tortuous and annoying language that normal people don’t want anything to do with. At some point you have to ask: is a book that 99% of the population will loathe and despise every single word of actually “great” or is it just that the literati are addicted to self abuse? You know my vote on that one.