There's plenty that's not actually perfectly understandable to a modern audience in his own words.
So long as that modern audience is sufficiently well-educated in one of the most important bodies of work in the English language (should this not be a major part of formal education? Else what is it even for?) there should be no problem. Therefore, if there is a problem then it is clearly with an education system that has apparently produced a nation of halfwits, rather than with one of the highlights of human achievement.
Yes, all we need is to be sufficiently well educated in the idiomatic usages of our language four centuries ago in a different country.
This sort of comment proves Al's point. Shakespeare is a cultural touchstone of the Anglophonic world that has inspired countless writers after him, and that goes all the way down to his poetic turns of phrase. A work of literature inevitably loses something in translation, so we shouldn't be translating something if we don't need to. The Iliad needs to be translated, since most of us can't read even a word of Ancient Greek. Shakespeare is similar enough to modern English that you can read the original text of his plays with a little training.