I don’t think we should start with something entirely abstract. Indeed, I want to start with something much more concrete: physical pictures of vectors as arrows, and a collection of concrete geometry problems to solve. If we later want to layer in a coordinate system (e.g. because it helps us solve one of our listed problems), we can draw several possible sets of coordinates over the top of our pictures, and think about how the computations in each coordinate system change while the underlying vector arithmetic stays the same.
Coordinates are an abstraction layered on top of vector spaces, and it is difficult for many students to understand the difference between the system or model being studied and the arbitrary details of one representation. I think focusing on coordinates narrows students viewpoint and leads them to a more rote/computational and less conceptual understanding. To the point that e.g. many undergraduate physics majors have difficulty with tests of basic competence about vector concepts, despite needing to solve a variety of hard computational problems.
Indeed, this specific failing of conventional formalisms is one of the primary reasons to switch to geometric algebra as a modeling tool in the first place.
It’s like trying to teach someone to play chess by looking at “algebraic notation” instead of starting by looking at a board and moving physical pieces around, or teaching someone phonology of a new language by looking at prose descriptions in their original language and static diagrams of a mouth, instead of starting by just directly listening to the sounds.
New languages, systems, abstractions, etc. are best learned through (at least some amount of) immersion in the mode of thinking/communication of the language, not purely through formal analysis using external tools. The presentation in the OP leads me to believe that the author (like myself, to be honest) is not actually fluent with thinking about geometric algebra concepts / tools, but is treating them like a novice second-language learner.
(Sorry if that sounds harsh. I’d love to buy the blog post author a coffee or beer if he’s ever in San Francisco.)
Though I have to admit not to know geometric algebra, from both you and the article, I gather it was created as an improvement on linear algebra. For those less innately interested in maths, starting from such a position tends to work much better than starting from scratch.
I feel like there are 2 methods aimed at different people. For the 'real' mathematicians, the approach you advocate is better suited. For those who use maths only as a tool, and do not so much appreciate the innate properties of maths, the easier to follow (but less informative) method of the author might be better.
Sadly, there is no single 'best method' for teaching a thing.
The article states at the beginning he want's to start with a concrete example, instead of diving into abstractions and formalisms. You respond by complaining this isn't abstracted enough.
I get your point, because this might set up some wrong intuitions. On the other hand though, the more abstract approach might be so dense as to resist any intuition at all. In my experience, wrong intuition is a better starting point for learning math that no intuition at all.