[gerdesj]

OP is able to create random points to infinite precision or the spherical cow has as many points on it as you like. Many here don't have the luxury of a Hilbert monitor.

If you convert (non mathematician here!) your sphere into an n-agon with an arbitrarily fine mesh of triangular faces, is the method described by OP still valid. ie generate ...

... now I come to think of it, you now have finite faces which are triangular and that leads to a mad fan of tetrahedrons and I am trying to use a cubic lattice to "simplify" finding a series of random faces. Well that's bollocks!

Number the faces algorithmically. Now you have a linear model of the "sphere". Generating random points is trivial. For a simple example take a D20 and roll another D20! Now, without toddling off to the limit, surely the expensive part becomes mapping the faces to points on a screen. However, the easy bit is now the random points on the model of a sphere.

When does a triangular mesh of faces as a model of a sphere become unworkable and treating a sphere instead as a series of points at a distance from a single point - with an arbitrary accuracy - become more or less useful?

I don't think that will be an issue for IT - its triangles all the way and the more the merrier. For the rest of the world I suspect you normally stick with geometry and hope your slide rule can cope.