Yeah, Shakespeare is not going to land if iambic pentameter doesn't sound like anything special to you. It's a big part of why many Shakespearean quotes stick so hard in people's brains, like 'To be or not to be', 'But soft, what light through yonder window breaks?' and 'If music be the food of love, play on'. If all of these have to carry their weight with only their literal words on the page, and not how they're actually said and stressed, they're going to be a lot less memorable.
Thank you for recognizing the situation I was in! (Nit: "to be or not to be" is probably a bad example of something that depends on iambic for memorability.)
Also, when I tried rewriting that soliloquy for modern ears, it didn't even occur to me to keep the meter; that never registered as a noteworthy aspect of it.
One other thing: keep in mind what it's like to be in high school. For the vast majority of your life, you're told to do things "because I said so". So the idea that there might be an actual good reason for stressing Shakespeare the right way ... didn't stand out as an obvious hypothesis.