I think most will tell you the axis of lateral pivot does matter, and that Cook is correct in regards to it. Where people disagree is in where he takes this discussion. If I remember correctly, his main argument surrounding the axis is for the shorter bit. Peaking at the book, this is due to 'wrist-pivot deflection error', which means that a longer bit results in greater cutting errors (less accuracy). But in another chapter he discusses the differences of curved and straight handles, and he is clearly opposed to curved handles as he says they effectively, at best, double the axis of lateral pivot. This would mean that accuracy would be twice as bad with a curved handle.
This is where people disagree. Some are quite opposed to this theory and say there is no difference. There was quite a long discussion on it, probably easy to find. In my thinking what he says is true, and this is proved in that for the same distance swing the curved-handled axe must be held at an angle which is equal to the fore-section. Or in other words, the curved-handled axe creates the same situation as increasing a straight-handled axe swing by the length of the fore-section. You can test this by comparing your wrist position on both handle-types, your wrist is always turned further back to the same degree as the curve in the handle, which effectively means that you can always get a longer swing for greater accuracy with a straight handle.
This may not seem like much, but in axework accuracy is king. One can quite easily begin doubling their time with missed swings. Of course, the advantage of the curved handle is that you can increase your power due to increased leverage. But if you do not pull the axe back further than you lose this whip advantage.
Note that my argument is one I developed on my own through the curved/straight discussion here on bladeforums. Cook's argument, or at least the one people take issue with, is that the curved handle effectively doubles the wrist-pivot deflection error. Their argument is that turning your wrist does not change the distance on a curved handle compared to a straight handle. But this argument misses the point, as Cook is not discussing intentional wrist deflection, but rather unintentional wrist deflection. And I feel that my discussion of wrist position at the starting point of equal distance swings is good proof of this: for an equal distance swing you are indeed moving your wrist the equivalent of a fore-section further.
Cook also discusses other shortcomings of curved handles, including the necessity of the wood finishing thicker and thereby losing some of the whip advantage of thin straight handles. This would mean that the whip generated from the extra lever in the design first has to make up for what is lost in the handle thickness. It gains only after it has lost in the first place.
