I had the same experience reading A Midsummer Night's Dream in high school -- I could follow the story, but it was a slog to get through and I didn't understand what was supposed to be so great about it.
Then I saw an actual performance of the play recently, and it all clicked. It was just as understandable as any contemporary film, and easily one of the most fun and entertaining stories I've seen.
I've had a similar experience, with the same play, but in an absolutely minimalist setting. No sets, the only props were a few chairs, and costumes were the same white slacks and smock worn by all the players.
And it absolutely worked.
This wasn't long after my secondary education --- I'd just started uni. But enough had happened that I got it.
As I've commented recently, even as an excellent reader, there's something that hits quite differently from hearing fiction (or even nonfiction) not just read but acted. Selected Shorts is an excellent source for this. See: <https://news.ycombinator.com/item?id=49163756>.
I think that is one thing about trying to study Shakespeare in class that can be improved. The class shouldn't just be taking turns saying the lines out loud. But you don't need a full theatrical production either or memorize lines. Read the lines with some light acting out of scenes with just the tables, chairs, and a small bit of imagination could be the class room study impactful.
I couldn't find the exact point that I was thinking of (when Hamlet is telling Ophelia that men are basically all just assholes), but I think this modern-day version of Hamlet helps to illustrate your point: https://www.youtube.com/watch?v=LdZVR4Ry3jQ
I gave up trying to read plays shortly after school. There's no point to me, they are supposed to be performed. I still think it's worth studying some of the more famous ones in school, but any study should absolutely involve actually going to see the play at some point.
I commented elsewhere but my schooling had a pattern of: read > analyze/discuss > watch movie > analyze/discuss. Most of the material has film adaptations and this was fairly sufficient in closing the loop during the course.
Even in history class, when we studied key documents the teacher would often at some point show us an oral presentation of the document- like a clip of (actor portrayed) Abraham Lincoln reading the Gettysburg Address. It’s such a well written speech that’s it’s exceedingly simple to comprehend when read. However, when you hear it orated it becomes evident how powerful it was and monumental moment in American history to justify learning about as it was unifying and motivating for all even in the midst of civil war. (Sorry may not be relatable reference for non Americans was just the first that came to mind)