For me, a lot of the un-cool-ness comes with being forced to read and often (over)analyze the books.
I adore The Hobbit and the LOTR trilogy and most of the other Tolkien books. BUT I was forced to read The Hobbit in 8th grade, and I hated it for the duration of the forced reading, and for quite a while thereafter.
And, I notably flunked one test because I couldn't tell the teacher what had happened in Chapter 12 - I had (re)read ahead so far that I didn't remember what was in Chapter 12 vs. 13 vs. 14...
I do think it's reasonable to have some reading lists that mean that kids will read widely across various genres and various types of "classicity", but still, for me, being compelled to read anything makes it a lot less enjoyable.