I believe that the whole timing thing for CF barrels came about as a result of the practice of indicating a barrel in, through the headstock, in such a manner so that the chamber end of the bore is coaxial with the rotational CL of the spindle. In order to arrive at this alignment, the position of the muzzle end can be significantly off off of spindle CL. At that point, the question arises as to how that offset should be clocked relative to the action, and the page, that I published a link to, (and the link within that page that pointed to similar work done with rimfire rifles) was included to point out that evidently that clocking makes a difference as far as accuracy is concerned.
It is my understanding that competitive barrels have been chambered by all three common methods, two through the headstock, and the third using a steady rest. I think that this is good news for those who have lathe issues that make it difficult or impossible to use one or more of these methods. A short bed could make using a steadyrest impossible for a given barrel length, just as a long headstock might make it difficult to indicate both ends of short barrel, or even secure it at the outboard end of the headstock.
Even if a barrel is indicated so that both of its ends are on spindle CL, if there is a curve between those points, the question might arise as to how that curve should be clocked relative to the action, even though the OD of the barrel, and the bore at both ends are centered.
When we pressurize a water hose that has a closed valve at its fee end, it "tries" to straighten out. I think that to a much lesser degree curved bores may do the same thing as they are fired, with the inertia and mass of the bullet causing the straightening pressure. It is my guess that the motion induced by this mechanism that causes differences in group sizes when barrels are clocked in various orientations to their actions, but this is just a guess.