Just to clarify a few facts:
Being educated doesn’t make a man a playwright, especially in the 16th–17th centuries, when there was no such thing as a creative writing class, and all university sessions were in Latin, not English.
Being a lawyer doesn’t make a man a great playwright.
Being a scholar doesn’t make a man a great playwright.
Bacon wrote some poetry, but none of any importance.
Stratford-on-Avon was certainly not a “town of illiterates”. For one thing, it had what modern Americans would call a “prep school”, which offered free scholarships to any student who needed one.
We don’t know for sure whether Shakespeare’s parents could write or not—there’s just no evidence—but his father was on the town council and had one term as High Bailiff—more or less the mayor.
We don’t know for sure about whether one of his daughters could write or not, but the other one definitely could.
There’s no reason to think he never wrote a letter. People didn’t generally save old letters back then, unless they were business, legal, or church documents that needed to be filed. And people didn’t collect autographs, either.
The “new” plays in the First Folio just hadn’t been printed before, which is not the same thing as being “new”. Plays generally didn’t get printed back then, unless they were old and not being performed anymore, except by pirates. When actors bought a play back then, they didn’t want other actors doing the same play for free, and there were no copyright laws, which weren’t invented until 1710.
I may seem to be going on at inordinate length, here, but people really do go down the rabbit hole with this crap, and some of them never find their way back to the human race again, wasting their lives spouting nonsense about conspiracies involving the Freemasons and the royal family and the Stratford-on-Avon Birthplace Trust and singing choruses of “Tomorrow Belongs to Me”. It isn’t pretty.