As I suspected, it IS possible to do this with Google Earth, but it is hardly obvious! After much finagling, I managed to figure out how to do it, but it is hardly obvious. I just managed to discover it by accident. I knew it was there, but I couldn't figure out how to enable it!
Here is the process (and "secret")
In GE, select the path that you want to edit (expand My Path so that it shows the path symbol and a number)
Here is the "secret". You have to right click on the Path and select "Properties". A window will pop up that says Google Earth - Edit Path (not Properties!).
Now each of the points that the path is constructed from will appear, and can be deleted, moved, or added to, etc.
See also http://bbs.keyhole.com/ubb/showflat.php/Cat/0/Number/515437/Main/274300