I love this question! Thanks for asking this because it really gets into the nitty-gritty of what we do as historians.

The most important thing to know in answering this question is that many historians (including myself) have what I like to call an "archaeological approach" to their work. Although we're not necessarily digging in the dirt and uncovering the structures and materials of human life from the past, we're doing similar work digging through archives to reveal the documentation of structural forces/processes and the materials of record-keeping that inform the creation, survival, and historical place of past culture and society. This is especially true of Shakespeare. Since the latter part of the twentieth century (largely thanks to the theoretical and historiographical influence of Foucault) there has been a growing emphasis on the world in which Shakespeare lived. Basically, it's historians asking "why did Shakespeare write the way he did" and what were his influences? The 2013 article you cite does much to explain the importance of knowing about what was happening in and around Shakespeare's life. As the article points out, such archival work helps us contextualize characters, plots, situations, speeches, etc. that appear in Shakespeare's plays and poetry. And we're actually quite lucky that Shakespeare lived Early Modern England as Queen Elizabeth I ramped up her government's oversight and documentation efforts as well as increased the size of the bureaucracy in an effort to modernize the English state and better root out Catholic dissenters. Because of this, the historical record created during Shakespeare's life is actually very large and requires a lot of time and effort to comb through looking for information not only about Shakespeare himself but the world around him. So we're still finding information that has long been placed away in an archive and has been overlooked by those more interested in studying the plays of Shakespeare but not his life. I would suggest you read/watch Michael Wood's In Search of Shakespeare (there is a book--cited below--and a 5-episode series). He makes use of a LOT of archival material to help piece together an image of Shakepeare's world and how we historians contextualize such information into a narrative about Shakespeare's life and his works.

But I would strongly caution against thinking that we know little about Shakespeare's life. As I mentioned, the English state under which he lived was highly suspect of its citizens and produced a massive amount of records that help us understand much about Shakespeare. Differences in spellings occur through these records because unlike our world today, spelling was not standardized by a national education system. Furthermore, dialects may have been much more pronounced, resulting in record-keepers writing words as they heard them spoken. Different spellings in a name may mean anything from different people to just scribal "error." Therefore, when we have a name that looks like Shakespeare we have to examine the record in which we found it and compare that to what else we might know about what was happening in Shakespeare's life that might lend credence to the man himself. We have to do the work of historians. And we know MUCH MORE than what that quote is suggesting because historians have done much to create a historical picture of the world of Shakespeare. For example, we can be certain that he is referred to by Robert Greene in the tract, Greene's Groats-Worth of Wit (1592) because in his description of an "upstart crow beautified with our feathers" features language that was common in Greene's references of London writers, like calling his work a "Shake-scene," and alluding directly to lines from Shakespeare's earliest plays. We also know Greene's feelings about his contemporary playwrights and poets and his affinity for giving them monikers that signaled their education and learned styles. He's clearly calling out a playwright who, unlike himself and his regarded peers, did not go to university (i.e. Shakespeare). There are many examples of evidence like this. The plain truth is this: can any historian say with 100% accuracy that a reference, play, or artifact is attached to the man from Stratford named William Shakespeare? No. But that's why we take the "archaeological approach." When we dig up something interesting we meticulously document all that we find. We ensure its historical veracity. We compare it to other similar artifacts found by other "archaeologists" in our field. And we publish or present our findings publicly to receive feedback and criticism on what we believe to be the proper historical contextualization for what we've dug up.

Shakespeare was Shakespeare. There is no other historical interpretation that fits. PERIOD.

Bibliography:

Michael Wood. Shakespeare. New York: Basic Books, 2003.

Alfred Harbafe. Shakespeare and the Rival Traditions. New York: Macmillan, 1952.

Holger Syme. Theatre and Testimony in Shakespeare's England: A Culture of Mediation. Cambrige: Cambridge University Press, 2012.

Jonathan Bate. The Genius of Shakespeare. New York: Oxford Univesity Press, 2008.

The New Oxford Shakespeare: Authorship Companion. Edited by Gary Taylor, Gabriel Egan. Oxford: Oxford University Press, 2017.

EDIT: I know I've glossed over the work of literary scholars, so if anyone care to comment with sources that promote the methodologies and analysis of literary scholars please do so!