In a very real sense, Shakespeare's plays have always been in contemporary settings. The stagecraft of the English Elizabethans put very little focus on setting or period. Actors' costumes varied more based on the class of the character they played, since working as an actor was one way around the sumptuary laws that rather carefully regulated the fabrics and colors people of different incomes could wear. While some costumes were likely purpose-made, many others were hand-me-downs from rich patrons. They just wore their own fancy clothes and trusted the words to establish setting. A close look at some of the plays makes this reasoning fairly clear - there is nothing particularly local to Messina in *Much Ado About Nothing*, and Shakespeare almost certainly never went there or anywhere else in Italy. He routinely gets geographical details wrong (Prospero refers to sailing from a landlocked Milan in *The Tempest* and Antigonus lands on the coast of a similarly landlocked Bohemia in *The Winter's Tale*). Setting plays in Italy was in fashion primarily because the Renaissance had spread from there first and claiming various Italian settings could give a kind of shorthand for the audience.

The Elizabethans weren't alone in wearing their own clothing. The next true superstar Shakespearean actor, David Garrick (1717-1779), wore clothes of his own period as well, though he threw in a trick wig for *Hamlet* that made his hair appear to stand on end when he saw the ghost. The first era I can think of seeing an actor in stereotypical Elizabethan dress is Edmund Kean (1787-1833) in the early Nineteenth Century, and remained in fashion throughout the Nineteenth and early Twentieth Century.

I would like to add a few elements of general context :

First, as someone else has noted, "historical accuracy" was not really a thing prior to the XIXth century. If you played a roman, you'd play it in with (fancy) clothes of your own time, the "setting" was not very important in terms of staging. One should also add that Shakespeare's plays were represented with minimal stage setting so there was no question of a "realistic" duplication of some foreign/antic world.

That started to change at the end of the XVIIIth century and begining of the XIXth. One of the main influence on that score was the romantic movement. Indeed, romantic poets and writers put a lot of emphasis on the "local color", the setting; they also had a strong interest for history. Novels, for example, would start to be much more descriptive when they dealt with history. That trend communicated to theatre : new plays written would start to include elements perceived as "typical" of the setting used, rather than treating it as a generic background that could be anywhere. The stage setting also became more specific. This was linked to the development of "staging" as an art form, and the rise of directors as a profession ; more and more, you had someone who was responsible for the general appearance of the production. (Which is important if you want an historically accurate production, because you have to get every actor the "right" costume, which was not really possible when everyone was responsible for their own clothes and no one was in charge of deciding what the general esthetic should be like). So, by the middle of the century, "historically accurate" productions of classical plays were starting to become a thing. (whether they were really accurate is an other question).

That trend, however, did not last long, because directors were still rising to prominence as a profession. Their first job had been to be in charge of the general visual appearance of a production. But, in the last decade of the XIXth century, their role was being, once more, redefined. More and more, they became responsible for giving a “point of view”, a reading, on the play. In that regard, a very important production is Gordon Craig’s 1911 staging of Hamlet at the Moscow Art Theatre. It was not done in “contemporary setting”, but it was not an historical one either. Rather, Gordon Craig choose a symbolist setting, where the stage setting was very abstract and supposed to represent Hamlet’s emotions, and the play main symbols. This is a massive event because, for one of the first time, you had 1) someone who had thought about what the general appearance of a play should be like 2) concluded that it should not be an historical reconstitution but rather tell a story to the audience today. Of course, Gordon Craig was not the only revolutionary director at that time (think Appia, Antoine, Irving…). But that staging in particular what very influent in showing that Shakespeare’s plays could be performed in any setting ; what mattered what the director artistic vision.

Th XXth century was really the century of directors, who all had to have a vision of the play they well willing to stage, something to tell with (and about) the play. So, of course, changing the context became a more and more accepted practice because it allowed for a very clear vision, a meaning. But that was not the only way a play could be “interpreted” : whereas in the centuries before, playing a part was largely a matter of tradition, now directors had a “point of view” on the roles (for example : Richard the third is really a victim, Lear is cruel at heart, and so forth). So what you say is the “modern staging” of play is really more of a symptom of the modern vision about what a staging should be : a reading of the play, almost a discourse on the play.