The original audience being illiterate is irrelevant, as it was being spoken in the language they were used to.
No, not really. More on this in a moment
Also, I hate this nonsense about "hidden meanings" that only come from pretentious professors reading it and re-reading it.
Depends on which "hidden" meanings you're talking about
Shakespeare was a PLAYWRITE, not a NOVELIST. The only people who were intended to read the text were actors who are studying the role for aspects that can be delivered via performance to a roomfull of people. They were intended to be viewed by people in an uninterupted environment (with possible intermission exception), so if the audience member can't get the meaning behind something when they can't stop, rewind and replay, then it isn't there.
I don't think you really have the foggiest clue about the world in which these plays were put on (oh and he was a play
wright, not a playwrite, he was also a poet). The meaning in Shakespeare that you think are hidden were fairly apparent, even to the illiterate groundlings. Shakespeare, and indeed other Elizabethan and Jacobean (not so much Caroline) playwrights, created some of the most reference dense works ever performed and the great thing is the audiences generally understood those references.
Just because you aren't up on mythology or theology or history and any of other categories of reference used in a play doesn't mean the fault in you not understanding them lies with the plays. Or, for that matter, with people letting you know what those references are.
Oh and Shakespeare was writing in, and indeed contributed to, early modern English, it's really not all that difficult at all except for spelling changes, it's hardly
Piers Plowman* for Pan's sake.
* Which is also only really difficult to understand if you convince yourself that it is.