I think that there are two cases of practical importance, which have incompatible requirements.
The first case is where you have a N-dimensional vector space where all dimensions have the same units. The standard example would be the Newtonian 3D space. Depending on what you are trying to do, you can view it as a collection of coordinate-free abstract vectors, as a triple (x, y, z) of real numbers, or as an array X[i] of three coordinates in a given basis. In this case I would agree that X[0], X[1], X[2] is better than (x, y, z), the order matters, and you can define the Jacobian is a 2D array that represents a certain abstract derivative in a given coordinate system. I would argue that the formalism of Sussman and Wisdom (which they got from Spivak) is totally adequate to this case, and perhaps even the best possible.
The second case is the one of the Lagrangian that parent mentioned, where L is a function of the triple (t, x, v). You could pretend that (t, x, v) form a vector space, but this definition won't get you far. I would regard (t, x, v) = t * (1, 0, 0) + x * (0, 1, 0) + v * (0, 0, 1) as meaningless because it is adding time, space, and velocity. You cannot really do rotations or general linear transformations in this space. You can define a Jacobian matrix if you want, but now all entries in the matrix have different physical units. In this case I would say the fact that v is the third element of the tuple is irrelevant, and that the tuple is better regarded as a map from symbolic names "t", "x", and "v" to real numbers. I would argue that the Spivak formalism is inadequate in this case, and it seems that many physicists on this thread think the same for essentially the same reason.
This difference is kind of analogous to double X[3]; vs struct { double t; double x; double v; }; From one point of view they are the same, but in practice they have totally different meanings.