What I don't get though is why less spokes mean more axial stiffness
It's not fewer spokes, it's fewer crossings. This increases axial stiffness by decreasing the vertex angle of the cone containing the spokes. A witch's hat is harder to crush vertically than a Chinese farmer's hat is.
Ouf, that exceeds my understanding.
What I don't get though is why less spokes mean more axial stiffness. Doesn't that depend on the hub width?