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. 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.
Notions of realism or naturalism, not our modern ones of course, but something ancestral to it was the trend of the whole period. In some respects the Elizabethan stage was a naturalistic revulsion against the morality play. At this time, prose dialogue instead of poetry was cutting edge naturalism. Moving away from stylized gestures, makeup and costuming to contemporary dress was a minor part of this perhaps.