I seek to provide a counter-proof against the following statement about the unit sphere in a N-dimensional space, with a large value for N.
Statement: Two randomly selected points on the surface of the sphere are nearly orthogonal.
Counter-proof using a random arc length
Create an arc of a unit circle with a length of random value in the interval of (0, π) and drop it on the surface of the N-dimensional sphere at a random place with a random orientation in the (N-1) dimensional surface. The two ends of the arc locate the two random points. However, the length of the arc does not have a high probability of being nearly π/2.
Causes for the orthogonality found in theory and in computations
I think the pole and equator method is only one of the multiple possible methods to estimate the probability in this case. Random selections in a multi-dimensional continuum might have indeterministic probabilities.
Question: What are the flaws in my reasoning?
This is not a counterproof.
Implicit in the original statement ("two randomly selected points on the surface of the sphere are nearly orthogonal") is that the randomly selected points are (separately) chosen uniformly. The fact that the conclusion doesn't hold for some other distribution you propose just shows that your distribution (on pairs of points) isn't the uniform one.
This is, by the way, the same phenomenon illustrated by the Bertrand Paradox. You might find it instructive to study that paradox, which is (at least dimensionally) simpler than the situation you describe.