The only way I was ever able to "edit" an alignment COGO point was using the lot layout tools. And as I recall, the conditions had to be just right. I think the thoughts behind alignment COGO points was that these COGO points were located using COGO and once assigned to an alignment, used the coordinates of those points. But if you edit the alignment such that the alignment no longer lines up with the points, they are discarded as alignment points.
The alternative is to convert alignment points to COGO points, edit the COGO points and redefine the alignment using the points.