I don't think you realize, Shakespeare was not from a family of "peasants." He was a family that had family that had strong middle class and entrepreneurial values. John Shakespeare desperately wanted to be a a gentleman with a coat of arms and he passed that desire on to his son. William Shakespeare felt he had succeeded in life because he had attained that, more than anything else. As for the education, it is only conjecture that his education was a poor as you say. He was removed from school at the age of fourteen for financial reasons, but what his education did consist of is a blank.
There is a possibility that his family were recusant Catholics, in which case, he may have had an excellent education on a more clandestine basis.
I just saw your post after I'd posted mine. I thought the same thing.