I recently read about a person who became skeptical of Shakespeare's authorship when he learned that there is no extant diary of Shakespeare's. He later became an Oxfordian—presumably on other grounds, because the Earl's diary isn't extant, either.
Modern expectations are often let down by Early Modern culture. It's similar to the Marlovian disappointment with the death entry for Shakespeare in Stratford-upon-Avon's burial record, which I've written about at the end of this post. The fact that Shakespeare's burial record didn't read "William Shakespeare, poet" does not mean that he was not a poet. The fact that we don't have Shakespeare's diary doesn't mean that he didn't write the plays attributed to him.
The list of great dramatists of the English Renaissance for whom no diary exists is expansive. We have no diary from Christopher Marlowe, Thomas Kyd, Thomas Middleton, Philip Massinger, Francis Beaumont, John Fletcher, or John Ford.
Ben Jonson comes closest to leaving us with a diary: his Timber: Or, Discoveries: Made upon Men and Matter: As They have flow'd out of his daily Readings; or had their refluxe to his peculiar Notion of the Times, published after his death, contains a considerable amount of autobiographical detail, but it's more like reflections on a life of varied experiences than a day-to-day account of his comings and goings.
Philip Henslowe's Diary is a chronological—and, often, a day-to-day— account of his theatrical dealings. It's wildly useful: without it, we would know much, much less about how theatres worked in the Renaissance. And I find it completely fascinating. But even this work—which bears the word "Diary" in its title—is not what the modern conception of a diary might imagine it to be. Knowing how much Henslowe paid "unto the nayllman for naylles" in 1593 (17) isn't exactly the sort of thing out of which gripping human-interest stories are made.
In short, however much we would all love to have a diary by Shakespeare—whether he kept one or not—it's not at all surprising or suspicious that there isn't one.