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 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 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.