Nothing has changed in the shooter's physical relationship to the balls, so there's no change in the "visual" he would have and no change in cut angle.
Here's a 2D picture that might help.

In this case, the translation between the two shots was done holding the ghost ball on the X axis, but that doesn't make any difference. The included angle between PG and PG' is about 15 degrees (or maybe 10 - I've forgotten - let's assume it's 15). For both shots, the player aligns themselves according to:
Cue ball center to left edge of object ball.
Right edge of cue ball to alignment point B on the object ball (i.e., the center of the object ball as seen by the player).
Consider the area bounded by PG and PG' both extended to the far rail. As I understand it, you are saying that this statement is false:
Line GC can be moved such that G falls on any point within that area without altering GC's orientation with respect to the table's X or Y axes.
Are you seriously willing to deny that? I don't think so.
Again, suppose we move A=(x,y) to a new point A'=(x+n,y+m), choosing n and m such that A' remains within the bounded area described above. This gives a new shot line, PA'.
Are you saying that we cannot move G,C as a unit, and without changing GC's orientation to the table's X,Y axes, to a point such that G lies on the extension of P,A' ? (For real balls on a real table, we must add the provision that C must have room to fit on the playing surface.) That would be equivalent to saying that a point on the X,Y plane cannot be moved to all other points on the plane by translation along the X and Y axes. Again, I really find it hard to believe that you're suggesting that.
The player's alignment is described identically for all shots within that area: CTE to OB left edge, CB right edge to OB point B, both taken as seen from the player's position. In order to maintain that visual alignment as the object ball is moved within that area, the player must physically move with respect to the table. This means that for all locations A(x,y) on which an object ball may be centered within that area, the player's orientation with respect to that object ball will differ from that used for some other location A(x+n,y+m), n,m > 0, unless A(x+n,y+m) lies on the line P,A(x,y).
So, do you still seriously contend that the alignment "CB center to OB left edge, CB right edge to OB point B, both taken as seen from the player's position" is not sufficient for an infinite number of cut angles within the 15 degree arc described above?
If you do, my previous premise was correct and the rest follows trivially.
If you do not agree, then it would be nice to see a drawing that illustrates some problem with the above, or get a coherent geometric explanation of where the above is wrong.