That’s curious, @Pachyderm_In_The_Room. I’d be tempted to look further. That sounds to me like one of those semi-plausible explanations that turn out to be conjecture. I would have thought that swearing oaths, especially oaths of office, would have been more common among the privileged than the criminal classes.
I would also have speculated that there’s a connection between the symbolic raising of the right hand to swear and the handedness of most of the population, hence not only associated with the use of weapons but also signifying something of maximum value. Think of gestures of loyalty, hand to heart, raising the right hand in affirmation, shaking hands, etc. (@SadieMartinPaul, I’ve read repeatedly that shaking right hands is a gesture of good faith, showing that there’s no weapon in your hand, but that too might just be attractive but unfounded lore.)
The reason we do it now, I would say, is custom. It doesn’t mean that anything is wrong with the left, any more than bowing in certain cultures means that something is wrong with not bowing in others, or covering the head in one means that it’s wrong to uncover the head in another. Consistency of practice in certain recognizable things characterizes a culture and doesn’t necessarily require an explanation.