Ugh! This perception is a lot of the reason people fail to see why the math is there for this, and continues to be repeated. Your statement, however, is incorrect. There are 2 variables for this reason. 1st, while the angle of the shot may remain the same, the angle going from center CB to center ghost ball (required spot to shoot center CB to make the shot), or even the CTE line WILL CHANGE with any CB movement away from the original spot. This will result in the sight lines putting your aim point at a slightly different spot, so you will not miss, at least not within a certain range. You will eventually, of course reach a position where you will need to use different sight lines, thus the 12 line system (6 for each cut direction). Also, as has been alluded to a number of times, any movement of the CB in relation to the OB will make the viewers perception of the OB size in relation to the CB change, thus, when you sight the same lines and pivot, their will be some variation in the resultant aim point from the one done from a different spot.
If you can show me anywhere on Stan's DVD where it suggests that the mechanical CTE method depends on the CB table position, I will gladly review it. I did not see anything on that video (or in the many other CTE descriptions) that suggests that it depends on anything other than CB-OB distance.