When someone thinks of an average road/street, they (probably) think of opposing lanes of traffic. Curbside would be right-side. Yes, there is a curb on the left-side of the street, but it is on the far side of the opposing lane.

Example: if I tell you to drive two blocks down a "normal" street and then pull over to the curb, you will most likely not pull over to the curb on the opposite side of the street.

Also, isn't "curbside" used when talking about valet services, which generally have you pull over to a curb on the right when meeting the valet? (I don't know, I don't have the opportunity very often)