That’s an interesting idea. The best class involving tensors I ever took was an introductory course on differential geometry, and I still think the coordinate-free approach of thinking of tensors as multilinear functions from some number of vectors to some other number of vectors (or a scalar) is great. Everything else just involves picking coordinates and figuring out where the numbers to :)

But I probably like abstractions like this more than most people.

Oddly, undergraduate physics also seems to be missing another, arguably even more fundamental, tensor: the moment of inertia. You can get quite far (in three dimensions, and only in three dimensions) by thinking of rotation as a vector. (Or a quaternion if that floats your boat.) But you can’t get very far by pretending that the moment of inertia is a scalar, and you get very confused very quickly if you treat it as three scalars in the magical coordinate system in which you can write it like that.