The values of the "putative function" do not depend on the chosen coordinate system.

This is the essence of notions like scalar, vector, tensor, that they do not depend on the chosen coordinate system.

Only their numeric representations associated with a chosen coordinate system do depend on that system.

If you compute some arbitrary functions of the numeric components of a tensor in a certain coordinate system, in most cases the array of numbers that composes the result will not be a tensor, precisely because the result will really be different in any other coordinate system, while a tensor must be invariant.

All physical laws are formulated only using various kinds of tensors, including vectors and scalars, precisely because they must be invariant at the choice of the coordinate system.

Here here! Functions do not depend on your choice of coordinates, only the components of tensors do! I think this is why it’s important to keep covariance and contravariance in mind. While tensor(fields) do not depend on coordinates intrinsically, the way we represent them when doing calculations most certainly does, and this is usefully characterized by co/contravariance.

"Here here!" is indeed topological. "Hear, hear!", on the other hand, is conversational.

I love when grammatical mistakes become unintentional puns

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.

Plus, as if tensor fields on Lorentz manifolds weren't already complicated enough, physicists aren't happy until they can write down some differential equations. So not only are you doing calculus, you're doing it on curved manifolds, with complicated tensor objects, in the context of partial differential equations, which - in the case of general relativity - are non-linear. It's okay to admit that all of this is a bit hard. Hell, as the article points out, Einstein himself had trouble understanding them.

There is a way of defining a vector space without an explicit basis(just as a set with an addition and a scalar multiplication). Similarly, there is a way of defining a vector bundle without choosing an explicit coordinates (as an abstract vector space, as defined above, which varies with the point in the space).

Just as a (p,q) tensor is a multilinear object related to a single vector space, a tensor field is a section of a tensor bundle associated to the vector bundle. (A section is just a function on the underlying space whose value at a point lies in the vector space above the point.)

Usually, the vector bundle relevant in physics is the tangent bundle of a 4-manifold.

This abstract way of defining tensors and tensor fields is manifestly invariant under coordinate changes, but it takes some machinery to set up. Whereas the 'numbers associated to each coordinate system which transform in a certain way' is more direct, but the rules can seem arbitrary at first sight. Also, maybe this approach can generalize to allow more transformation rules which might take some time to put into an abstract setting.

Standard example is a matrix A which transforms as PAP^-1 (where P is linear coordinate change) vs a matrix T which is a linear map between vector spaces.

The same issue appears in software where you can expose a data structure as a tuple of numbers/string fields and then define functions on them, or you can expose it as an abstract data type where the user of the library can only apply certain functions on them and the implementation author can choose different representations(coordinate changes) in which to easily compute the functions.