I think what people have said here about returning to the classics when you aren't studying them is very helpful. I loathed English class, HATED reading what was given to me, I took as few University literature classes as possible. Why? It wasn't that I didn't like the classics. I'd loved Shakespeare ever since I was 9 years old (I remember my first encounter with it, being totally mesmerised by the musicality of the language despite not knowing what it meant). But I always hated analysing things I was reading. I wanted to immerse myself in the story, not stop and start. Further I think I always assumed I didn't understand what I was reading as we always analysed things to pieces. That if I just read such a work on my own I wouldn't get it without the help of a teacher so what was the point in reading it for enjoyment.
I'm glad I avoided English Lit, just for my own sake. It meant that when I came to the classics on my own, I could read them enjoy them and take from them what I wanted. Certainly now I analyse things more, but as I wish to analyse them.
I read The Great Gatsby this year for the first time. Absolutely fell in love with it. I remember people having to read it in high school, I never did. I doubt I ever would have read it again now had I read it then.
There is something to the classics. They last for a reason, and not just because the establishment tells us they rock. You might not like all of them, but the fact that you don't like any of them suggests to me that you've been prejudiced in your studies to think them inaccessible and old fashioned. The fact that all you see in Shakespeare is flowery prose when I see a writer of striking modernity is a good example of this (I mean, "I live with bread like you, feel want, taste grief, need friends" is a pretty common thing isn't it? The fact too that Shakespeare ends the line with "needs friends" is pretty something too. In writing we tend to build up to the biggest thing like: "He was the most popular guy in his class, his year, his whole darn school." So to have Shakespeare use friends as one of the biggest needs for this King . . . what could be more modern than that?).
In any event, no, you don't have to love the "classics" (whatever that even means), but you do have to read them. Because we as authors need to know what came before to understand where we can go. It isn't about intelligence. Your friends may be total geniuses without having read a single word. But this isn't about them. It's about writers. And as a writer, you need to know your craft.