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.