That's the thing, the number of new cars using that road ends up being at least one additional lane's worth. So traffic moves at the same speed as it was before the extra lane, just now with one more lane's worth of cars on that road.

If anything, you might see marginally better traffic on other roads because of the cars that started using the new lane, but you'd be talking about a handful of cars per road. Probably not enough for any discernible change in travel time or congestion, and each new lane you add later will have diminishing returns because it will be a smaller fraction of the total number of lanes coming from any specific direction.