I think this is far too simplistic, for one because the values of this putative function depend on the chosen coordinate system.
So I completely agree with the comment you are replying to: when a physicist says "tensor" they really mean a "tensor field" and the definition of the latter is quite a bit more involved than just specifying a multilinear map at each point of a manifold.
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.
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.
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.