Axial angular momentum calculation

[A.T.]

In the context of Newtonian physics I believe there is no gain from the bi-vector language/definition.

This is probably true for textbook problems, where simple systems are meant to be solved by hand. But this assessment might change in the context of automated computation of more complex systems, which have a lot of symmetry, like some robots or animated characters. Here, you often have to mirror the mechanics from one side to the other. And not having to keep track which vectors are pseudo-vectors, and thus have to be negated after mirroring, might simplify the code.

Think about how quaternions where replaced in physics by vector analysis in the late 19th century, but then had a comeback in the late 20th century in computer graphics and robotics:
https://en.wikipedia.org/wiki/Quaternion#History