This isn't a specific Russian problem. English speaking school children are forced to read Shakespeare, and I really don't think that works either. (That isn't a condemnation of Shakespeare but of schooling.)
I do love literature, but that is in spite of school not because of it. School did a lot to put me off some books. I was lucky to have read Golding's "Lord of the Flies" before our class did, because it gave me a better appreciation of it. I did read some big books as a teenager. I waited until my twenties to tackle Dostoyevsky though. "The Brothers Karamazov" was especially difficult.
Shakespeare is good for kids, its mostly quite light and fun and not very long, theres a linguistic challenge but thats a good learning opportunity
Shakespeare would land much better if people were reading it in a language they speak, as opposed to a language that he spoke.
When 90% of your mental effort is dedicated to understanding exactly what the hell he is saying, you aren't going to get a lot out of his work.
(It's not supposed to be read at all, in fact - it's supposed to be seen and heard. In a language that you intuitively understand.)
If your school just had you silently reading Shakespeare they were doing it wrong. It is meant to be performed and watched, his works are plays and poetry not novels. I was lucky, my English Literature teacher in high school was a (very) minor playwright and well aware of how important speaking the lines out loud is, and how watching a play is so very different from reading it.