An important thing to consider in evaluating what the fact that the Elizabethan/Jacobean theatre used contemporary dress for historical characters, and other anachronisms:
Theatre at this point did not have a tradition of Realism/Naturalism. There was no sense that the stage should try to accurately re-create real life. Sets, costumes, staging, characterizations -- these facets of the theatrical experience were not designed in Shakespeare's era to be an accurate reproduction of the physical world, any more than music is meant to be an accurate reproduction of human speech or of the aural world.
So the idea that there would have been any need or obligation to use period accurate dress, or to refer to accurate technology, for historical plays, just would not have occurred to them.
I find this more plausible than the certainty that Shakespeare was modernizing the stories by setting them in contemporary dress. But I have been authoritatively assured that this was so.
I suppose one might argue that the effect
of using contemporary garb was to "modernize" historical characters, even if this was not the intent. But I think it's far more plausible to presume that theatrical companies of the era just didn't put that much thought into it one way or the other, given their lack of a Realist/Naturalist tradition. They just did not care about anachronisms.
Which, either way, does make the presence of anachronisms appropriate for a Shakespeare adaptation set in modern times. These anachronisms might not have registered as such for an Elizabethan audience, but they were certainly present, and it seems quite appropriate that we can encounter anachronisms our current productions, if only to "mirror" the anachronisms present in the original productions.
Further, the reverence for Shakespeare is most commonly held to be precisely his realism about humanity, albeit of a timeless variety. I don't actually believe in a timeless humanity, so the reverence is confusing.
*shrugs* I like Shakespeare a lot; I think he was a great writer with amazing insights into humanity. I also think he was a man of his age, and that the prejudices of his era clearly show in his work. I also think he was a popular entertainer who lived in a dictatorial police state, and that sometimes his works functioned as political propaganda.
I think that Shakespeare may be an amazing writer, but that we in the English-speaking world have a habit of setting him up on a pedestal and engaging in "Bardolotry." Yes, this is the man who wrote the sophisticated characters of Hamlet and Macbeth; this is also the man who created the ridiculous cartoon characters of Pistol, Nym, and Bardolph from Henry V
, whose sexism shines throughout The Taming of the Shrew
, whose idea of a typical working-class man can be seen in The Tempest
's Stephano or in The Winter's Tale
's Clown and Autolycus. The same man who wrote "To be or not to be" was not above dirty jokes about the word "cunt" in Henry V
, or above absurd physical comedy in Midsummer Night's Dream
Shakespeare was amazing, but he wasn't this perfect, idealized writer people think of him as. Were he alive today, the guy would probably be working in TV or movies, making stuff that alternates between easy laughs and sophistication.... sort of like Joss Whedon.