I agree with the comments that the way Shakespeare is taught at GCSE and A' Level is a big part of the problem. It's rarely if ever taught in any kind of context, with the result that Shakespeare is seen as 'stand alone' - which, of course, no author is: they and their work are products of the culture from where they originate. Shakespeare, J B Priestley and Steinbeck: what, exactly, do they have to do with each other? Where can meaningful links and distinctions be made?
There's no criticality. Othello and Merchant have definite shades of racism and anti-semitism. 'Taming of the Shrew, by today's standards, is sexist. A Midsummer Night's Dream is completely kooky and a hoot from start to finish, with definite links to the epic tradition. How do you square contemporary attitudes on these issues with those of today? Also highbrow -v- pop culture: Shakespeare is Take That compared with Marlowe's Beethoven. These are some of the questions the syllabus should be asking, at least in some kind of straightforward form.
The National Curriculum is unfit for purpose IMO and the texts being taught today are exactly the same ones that were being taught in 1988, when the GCSE was first introduced. Where is the work on refreshing that curriculum, of making it more relevant and fun? I suspect where: it's all gone into politicking. If I'm honest, I'm not surprised people have retained a dislike for Shakespeare as a result.
When I studied Elizabethan undergraduate modules on the Italian Renaissance tradition and its later English counterpart, the whole period came to life for me. The sonnets on their own are a fascinating story with major stories to tell about English culture. I'll never be a Renaissance expert, but I know enough to teach it at undergraduate level and it will always remain a pleasure rather than a specialism.