I think it takes some experience in getting a Shakespeare text from the page into a form that really clicks with people. It's not because the words are unfamiliar, but because Shakespeare has a fondness for long, elliptical sentences and convoluted grammar. It can be hard to figure out where the emphasis goes.
It helps that the poetry gives you a lot of cues. Iambic pentameter gives you a framework. Deviations from it are cues that something is important. It's not a formula for performance, but it reveals just how much more Shakespeare was directing you than just the words to say.
John Barton's "Playing Shakespeare" course has been how they teach that to RSC actors, and that has given us some of the greatest Shakespeare lights of the last 50 years. The videos of it with his students are magnificent views of a comically young Patrick Stewart, a ridiculously baby-faced Judi Dench, a positively embryonic Ian McKellen, and many others.
Inexperienced readers may not be able to get that just from reading it aloud by themselves. But if they listen to somebody who has practiced it, they'll have a huge leg up on comprehension.
