It didn't claim to predict the list very well. Its purpose is to give intuition about how far around the Earth we can see.
Also it shows that simply being tall (airplanes go higher than Everest) doesn't get you on the list. The second paragraph puts an upper bound on sight lines between two mountains of a given size. The longest sight lines are pretty close to that limit. And it gives a simple experiment that anyone can carry out from an airplane which will let you see the curvature of the Earth.
as its not a list of just the highest peaks in the world
longest sightlines also depend on the surrounding topography that blocks or doesn't block a sightline
this seems to account for the numerous Spanish entries on the list, where the large Spanish plain with high peaks on either side, enables you to see from 1 peak to the other, w/o anything on the plain blocking your view