Will a professor in his ivory tower hold up a copy of 50 Shades of Grey to the class of 2214 and proclaim, "THIS is when they wrote great literature!"?
One of the criticisms leveled against Danielle Steel is that she's a mercenary writer. That same criticism was leveled against Dickens, who's books are now classics, 150 years ago.
Neither Chaucer nor Shakespeare had much in the way of classics to be forced upon them but they both did OK with their works. So to answer the question regarding the importance of reading classics, it is not important - but you will attempt to read them anyway out of your own curiosity to determine why somebody labeled a particular work a classic. I can't for the life of me understand why The Count of Monte Cristo is a classic, I couldn't finish it after three attempts to do so. (But one of the difficulties I had with the book was giggling every time I came across "Danglers".)
Some adult, I don't remember who, gave me a slew of cheaply printed and bound paperback classics when I was a child: White Fang, Silas Marner, Catcher in the Rye and others. White Fang I read so many times that it fell apart, Catcher in the Rye fell apart because I flung it against the wall (Holden Caulfield was nothing but a whiny punk, I wanted to punch his face).
When someone quotes from classic literature, I feel ignorant if I don't know where the quote came from and in what context it was originally written. But rather than take the hours and days necessary to read an entire classic book, I just pop a few keywords into Google or Wikipedia and feel classically literate without actually taking the time to suffer through aged prose that barely makes sense today.