Comment by kevindamm
The first time I learned it was from a book by LaMothe in the 90s and it starts with your demonstration of 3D matrix transforms, then goes "ha! gimbal lock" then shows 4D transforms and the extension to projection transforms, and from there you just have an abstraction of your world coordinate transform and your camera transform(s) and most everything else becomes vectors. I think it's probably the best way to teach it, with some 2D work leading into it as you suggest. It also sets up well for how most modern game dev platforms deal with coordinates.
Think OpenGL used all those 2,3,4D critters at API level. It must be very hardware friendly to reduce your pipeline to matrix product. Also your scene graph (tree) is just this, you attach relative rotations and translations to graph nodes. You push your mesh (stream of triangles) at tree nodes, and composition of relative transforms up to the root is matrix product (or was the inverse?) that transform the meshes that go to the pipeline. For instance character skeletons are scene subgraphs, bones have translations, articulations have rotations. That's why it is so convenient to have rotations and translations in a common representation, and a linear one (4D matrix) is super. All this excluding materials, textures, and so on, I mean.