I thin you are right (and this is reflected in other comments) that some people find Elizabethan language hard and others do not.
There are lots of things that prevent some English speakers understanding others. Dialect, vocabulary and culture for a start. I often hear or read (and use) words in some places that I would not use in others (a huge range from lakh to orthogonal). I often read things that other people might have a problem reading, or things that contain words I need to look up.
Short of using only very simple language all the time it is impossible to find anything that is not going to present difficulties to some significant group of people.
> Despite these qualifications, I was hard-pressed to find any lines that I could make sense of in Shakespeare
Really. Take a look at the first few exchanges in the play I know best:
https://shakespeare.mit.edu/midsummer/full.html
The word I can find that has had a change of meaning is "revenue" and that is slight. You can read a long way down without seeing anything that is hard to understand.
My children have been able to enjoy Shakespeare from before before they were teenagers. I think it is because they read a variety of books so are not confused by unfamiliar language.