The "controversy" is a flawed fiction, a scurrilous invention utterly without any scrap of merit. There is literally no anti-Stratfordian argument that is not based either in logical fallacies, in grievous misinformation, or in outright lies. Not a one. They all stem from a lack of knowledge of early modern London, and of the theatrical and publishing culture of the times. Small wonder, then, that absolutely no one questioned Shakespeare's identity until more than two hundred years after his death.
The anti-Stratfordian arguments are also elitist and condescending. The foundation of their theory and one of their most cherished fallacies is that a man without a university education, a man who was not a member of the royal entourage, could not possibly have created such works of genius. I don't know what inspirational message people are meant to take from that—that you can only be great if you're born to greatness? That without advantages and inherited privilege, you'll never succeed? What snobbery.
As for the depictions of court life in the plays, it's hardly suspicious that the son of an alderman, who had connections through his company's patrons to the royal court and who could pick up a book on courtly manners (and scandals) or on English history at any Fleet Street bookshop, would know how to portray the upper classes. I could imagine that far more readily than that a nobleman could create the incredible pathos of Michael Williams, Alexander Court, and John Bates, sitting in the mud on a freezing night, discussing what common men sacrifice for great men's ambitions (Henry V).
William Shakespeare, the man from Stratford, wrote the plays we attribute to him. We know who he is. With no legitimate evidence to suggest otherwise, I can't imagine why it needs proving, but that doesn't matter—there's plenty of proof for anyone who cares to look. It comes from the words of his contemporaries: from historian Francis Meres, from the government officials responsible for his family's coat of arms, and from his fellow playwrights, from Jonson, Beaumont, Heywood, Webster, and more, all affirming his identity (even when, as was sometimes the case with Jonson, they did so with criticism, not praise). And it comes from his plays. The man who wrote them was someone who knew the theatrical world intimately, who knew the advantages and challenges of the Theatre and the Globe and the Blackfriars, and who wrote his plays with those conditions in mind. He was a man who knew the actors who would be playing for him, who knew that he could write a Dogberry for Kempe but that he needed a Feste for Armin, who knew he could trust Burbage with Hamlet and Othello. His stagecraft is the mastery of an intimate, not a visitor, and for me, that proves him past any question.