I want to know whether or not it would be perfectly legal for a US citizen to carry around a battle sword and scabbard today. If not, when did this law change and why?
There was no historical moment when open carrying swords was made illegal across the US.
Instead, the answer varies from state to state, and is a thus more of a legal rather than historical question. In my state, for example, open carry of swords is allowed, so long as no one feels threatened; but you should really discuss this with a lawyer rather than a group of historians.