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.