The author is missing a huge very real advantage of quaternions, that they are smooth functions of thier parameters and don't suffer from gymbal lock singularities. All 3d representions do, as can be shown with the math this guy is too lazybrained to do.
That really matters for defining smooth paths, quaternions simply have the right topology and rotations don't.
The author is using geometric algebra to create mathematical objects with all the same properties as quaternions, but to explain and understand them in a more comprehensive, sensible way. These things have all the same rotational properties as quaternions.
In particular, he’s explaining that quaternions are not vectors, they’re actually oriented planes and showing how their multiplication rules arise in a way that’s not ‘out of thin air’.
That really matters for defining smooth paths, quaternions simply have the right topology and rotations don't.