Perhaps Shakespeare was a man of little education, but, I don't recall that Sam Clemens had a college degree. Art can't be learned, IMHO, you are born with it; it is in-bred.
All the educated individuals in the world have not produced anything like Shakespeare, so, logically, one would assume that a lot of education is not relevant to the question.
No one has a problem attributing the works of Van Gogh, Clemens, London, the Beatles, etc. to the respective working-class artists for one simple reason: their works where known and their genius widely acclaimed (with the exception of Van Gogh) during their lifetimes.
The usual reason stated by Stratfordians why this isn't the case with their man is that these incredible poems/plays, obviously written for the ages (and recognized as such by the author himself), where *not* art; rather, they were merely the equivalent of TV fare for its day. To wit, who knows who wrote a series of scripts for a cancelled TV show?
If you reject that theory, you reject Stratford. If you read Shakespeare with the knowledge that the author knew exactly what he was doing at the time they were written, and that his genius *was* recognized furing his lifetime by those that had the ability to read & understand the many, many veiled references (ie members of the nobility & court), then you know that the name "Shakespeare" was merely a nom de plume.
Funny you should mention Samuel Clemens, because he is a prominent Shakespeare skeptic who wrote Is Shakespeare Dead?