Well it's almost impossible to anticipate every change scenario. It would be nice if we could just copy the labels as you are wanting to do, but that is not how it's designed. However you do have the ability to clone an existing label and apply it as needed. So in the scenario you presented, is it any more work to move the existing labels vs. cloning one and applying it to the new features? I'd choose the latter because it just may be that future changes to the geometry won't require new labels.
FYI, I also use Civil 3D and it's spot elevation labels work more the way you are wanting it to in Geopak. While GeoPak labels aren't nearly as robust, I do like the fact that they are capable of maintaining association with an element. Consider the case where you've labeled a curb elevation. That curb could move so slightly that you may not even notice that the label fell off the face of the curb and was pointing to the gutter. It would be totally wrong in C3D because the label isn't associated with the curb. A Geopak label would update to the new location of the curb.
I'm not enthralled with Labeler but it does have it's strengths in the scenario you are presenting.