Short answer: yes.
A few years ago, as a new retiree, I took a class in dramatic literature at a local community college. I was probably the only person in the class who was old enough to drink, and certainly the only one with a four-year college degree and a completed English major behind me. I took it just for fun.
Some of the kids were fairly bright, but I wouldn’t say there were any shining lights. In general these were pretty average youngsters going to a two-year college that anyone could get into by paying the modest fees.
The instructor started off pretty easy, with some very short contemporary plays, and worked up to Shakespeare. His choice was The Taming of the Shrew.
As background he included an in-class showing of a scene or two from the Burton-Taylor film version (1967). He also referred us to the movie 10 Things I Hate About You (1999), which is a modern teen-oriented adaptation.
One feature of the class was that the instructor used online media as well as traditional materials. Everyone had to post responses to readings online and then respond to others’ posts. I got to see many young people’s comments on the plays.
And I was frankly amazed at how much these young and inexperienced readers got out of this well-known Shakespearean comedy. Even granting that some of them may have had help with their essays, I heard their in-class comments. They got it, they got into it, they enjoyed it, and they were able to discuss it.
Remember that Shakespeare was written for a common audience in the pits and not just the swells in the boxes. The language is old—English has changed a lot in these few centuries—but not inherently too fancy for ordinary folks.