Shakespeare remains high on the list of influential writers, even though he sort of hid in plain sight when he was alive (some say it was because Shakespeare was not Shakespeare, but some other guy, but more modern explanation makes sense: he was sort of hiding in plain sight because he was a gasp Papist).
David Warren has an essay on him, noting (omong other observations) that one reason people never read him is because they were forced to read him in school.
That didn't bother most of my classmates, who promptly bought "classic comics" version, and managed to pass the tests without blinking an eye.
The reason is that Shakespeare was a popular writer. He didn't just cater to the intelligencia but to the folks who can and did buy tickets. So lots of romance and gore and humor in his plays.
Asimov has a short story where a physics professor discussed time travel and Shakespeare with an English professor...turns out that Shakespeare was brought to modern times, attended class out of curiosity, and flunked the course.
No, Shakespear would fit right in with the Hollywood hacks.
So what is the trouble with modern folks enjoying his plays? The language has changed.
Even if you watch them in a movie, following what is going on is almost like watching a movie with subtitles: Your mind has to summarize what the people are saying so you can follow the plot.
There is an argument that they need to be tranlated into modern English.
There are several modern versions of his plays that are watchable, but only as one watches a "classic", not as one watches a hit movie.
Ironically, a lot of Shakespeare's plots have been morphed into hit movies. The Taming of the Shrew, with Elizabeth Taylor comes to mind.
But too often, the films that try to "stay true to the plays" have the flaw that we see in "religious" movies: The people act like they are performing in a church service, not acting like real people.
Maybe this is why I prefer Shakespeare in the original Japanese : a pun on this famous line:
My favorite is Kurosawa's Ran, a remake of King Lear. It is one of the films I brought with me to the Philippines.
He also did Throne of Blood, a remake of Macbeth.
But modern versions, from "Ten Things I hate about you" (loosely based on the Taming of the Shrew) to West Side Story (Romeo and Juliet) show that the plots are still there for others to build on, for people to enjoy, not to force students to study because it is good for them, like medicine.