As weight is applied to the center of the wheel, the tire begins to flatten where it contacts the road. If there is no friction between the tire and the road, there is no external tangential force acting on the tire. So there is nothing that will cause the tire to change its length. The tire will change its shape, but its total length will remain the same. For this case, the effective radius is equal to the original radius.
In the other extreme, where friction force is very high, as soon as a point on the tire surface touches the ground it "sticks" to it. In this case, after the entire weight has been applied the tire looks like Figure 1. Because each point has "stuck" to the ground, the tire material in the arclength "s" in Figure 1 has been compressed to length "c."
Eventually every arclength "s" of the tire will be compressed to a length "c" while it is in contact with the ground, so that the effective perimeter will be given by
and therefore
Returning to the equation given by the motorcycle tire paper.
Effective radius = R-(R-H)/3
and replacing H with the expression from Figure 1.
I'm not too good at trigonometric identities, so at this point you will need to do what I did, plug in different values of alpha (make sure to use radians, not degrees) to see that both the expressions for effective radius, the one I derived and the one given in the motorcycle paper, give the same values, i.e.,
So it does appear that for the case with high friction and no slip in the contact patch, the motorcycle paper does indeed give the correct value for effective radius.