Arguably the reason that generations of STEM students have been horribly confused about 3-dimensional vectors and rotations (including electric/magnetic fields), etc. is that they were reframed in the confused and non-generalizable Gibbs/Heaviside language, instead of in Grassmann/Clifford’s formalism in which vectors and bivectors can be properly described as separate types of objects.
Geometric algebra obscures the point that what really matters is the algebraic structure. There are many ways to construct things that behave the same way (subgroups of matrices, vectors and the cross product, and so on), and while it is nice to invent one construction that gives you everything, you risk loosing sight of the fact that each construction is arbitrary and what really matters is its algebraic structure. That's why it's good to expose students to many partial, fragmented devices, so that they will realize the deeper point that underlies them all. With spin matrices, it is obvious that they are arbitrary manifestations of a group, but if a student were to spend their entire education manipulating blades they might start getting the idea that in some sense the universe was "made out of them."
Reminds me, I've been meaning to read this [0] blog post for a while now. (See also the HN discussion [1].) My clueless intuition tells me its points may be analogous to your linked document. (It mentions Heaviside, at least.)
It can be so much nicer. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf