You probably already have this in mind, but I think a cool figure here would be a visualization of all pairs of floating point numbers interpreted as stereographic coordinates put back on a sphere. (I suppose you'd have to use a much lower precision; apparently there are things called "minifloats"...)

Jacob pointed out this plot in another notebook of his, which has essentially what I wanted to see: https://observablehq.com/@jrus/pj#plot_pj11_fp11

After a year away I got back to fiddling a bit with observablehq.com/@jrus/spherepoints

Here is a better picture of quantized stereographic coordinates, observablehq.com/@jrus/quantizing-stereographic-coordinates

Another visualization idea: there's 1 parameter family of projections interpolating between the tangent and stereographic projections, given by moving the point of projection towards / away from the plane, cf. https://en.wikipedia.org/wiki/Gnomonic_projection#/media/File:Comparison_azimuthal_projections.svg . (There might already be some interactive toy that shows this that I haven't seen...)

Could be worth pointing out that Möbius transformations are precisely the conformal self-maps of the sphere, given the mention of a few other conformal things on the page. Also, I feel like linking the "Möbius transformations revealed" video is almost obligatory anytime they're mentioned, just because it's so well-done: https://www.youtube.com/watch?v=0z1fIsUNhO4 although maybe it ends up distracting in this context.

I didn't check the Stuart reference, but my understanding is that the Fubini-Study metric is a particularly nice metric defined on complex projective spaces (spaces of complex lines in a complex vector space). In the case of the 2-sphere (which can be viewed as the space of complex lines in C^2), the metric is (up to a rescaling factor) the usual distance on a round sphere, so I don't think it is the stereographic distance you've defined here.