But that just re-raises the question of why people switched to driving on the right. The article says riders used to ride on the left, then wagon drivers started sitting on the left, so wagons started driving on the right. Why did wagon drivers started sitting on the left, in a world where people were riding on the left?
For context, at this time in history, the side of the road used was mostly determined by custom and convenience - there wasn't enough traffic for there to be established law.
In the case of these large wagon train drivers, the drivers, being mostly right handed, wanted to use their crop or whip in their right hand while being able to reach both lines of horses. This dictated that riding on the leftmost horse was most convenient.
Conversely, a wagon driver in a (relatively) cramped and crowded city such as London would sit on the right side so as to be able to ensure that his whip did not accidentally hit passers-by, as the direction of travel of the whip from a right-handed person is toward the left. This concern was largely not present in the wide-open spaces of the US.
Wanting your whippin' arm lined up center makes sense, but it doesn't explain swapping both the side the driver sits on and the side of the road used by the vehicle. If you had free reign on street side, you'd sit on the side opposite your whipping arm because you could pick people up on either side (no safety preference) and would have more control over the whip. These probably changed at different times.
If you're on the left side of the street, picking passengers up on the left sidewalk, you'd want to sit on the left side to watch your whip. But, the driver is always on the side furthest from the curb, which makes me think it's more about making sure you don't hit oncoming traffic and picking up passengers.
Still doesn't explain the side of the street used. I wouldn't be surprised if there was a royal mannerism edict that said something like, "Gentlemen ambling in opposite directions should always pass one another on the left to ensure they may unsheathe their weapon, should the other engage in dishonorable combat."