In the end tensor fields are sections of a bundle. I insist that calling them "just functions" is simplistic. In fact, I'd say that the complexity of your elab

In the end tensor fields are sections of a bundle. I insist that calling them "just functions" is simplistic. In fact, I'd say that the complexity of your elaborations kind of proves my point. Note that I deliberately use "simplistic" and not "wrong", since a section is a function of sorts.