What clicked for me was after being forced to read The Merchant of Venice in grade 9, I suddenly saw all the cultural references ("I want my pound of flesh" or "All that glisters is not gold") in movies, music, etc. I remember watching Strange Brew and somebody mentioned that it's basically Hamlet and you can't unsee it once you realize which movie character is matched to which Shakespeare character.
Over the years I've been to several adaptations of A Midsummer Night's Dream and it was a ton of fun every single time.
Funnily enough I never read the translations, so can't speak as to how accurate your affirmation is, but I can see it happening for sure. I'd say it depends on the style of the translation. I've seen some try too hard to preserve the tone of the original.
We went through it, a bit at a time, in class. With a very good teacher. Not just helping with word and phrases, but digging in a little bit. And at the end we watched a performance of it.
I think that made it so much easier to appreciate, and it made it a ton easier to read and appreciate Julius Caesar or whichever ones we did next.
And like… that was 30+ years ago for me, but I feel like I see echoes of that stuff all the time.
I feel like in a way we're losing Shakespeare's intended effect, because we insist on using a "language" that isn't used anymore. Yes, I understand that it's considered the same language, but a language isn't really one thing, it's more of a spectrum without sharp edges. Shakespeare is much closer in time to Middle English (which is considered a different language) than he is to our time
Not to say it will be a complete failure, but it does lose some of its poetry and power.
And smut. There's a lot of smut and double entendre in Shakespeare. An innocent modern word like "nothing" has multiple layers of meaning in Elizabethan English.
Modern audiences don't get it unless someone explains it to them.
The net result is that it's easier (for me) to understand a letter written by someone in a translated foreign language than one written by someone in English from the same period.