You might be interested in geometric algebra. As Anthony alluded to, physicists are too often obsessed with expressing all geometrical quantities in terms of coordinates and coordinate bases. The more mathematical approach of putting everything in a coordinate-independent way helps, but this is usually done by considering everything as a map. You can absolutely do this, but it also tends to lose the geometric significance of objects. As an example, I spoke with a professor I had for a Riemannian geometry course, and I explained to him the concept that the Riemann tensor is a map from bivectors to bivectors (from oriented planes to oriented planes). He seemed positively surprised by this notion, but in geometric algebra, it is the only way the Riemann tensor is presented (and the only way it needs to be presented).
This is what traditional mathematical presentations lack--the easy ability to deal with geometrical objects beyond vectors, even though such concepts would bring considerable clarity to differential geometry.
One formalism of geometric algebra applied to differential geometry is the "universal" geometric algebra--an infinite-dimensional clifford algebra. This takes away the arbitrary nature of an embedding; a manifold can simply be considered as a set of vectors (not tangent vectors, but vectors in a flat space). And because the points on the manifold are themselves vectors, expressions like $\partial x/\partial x^i$ are no longer hand-waving nonsense but legitimate, well-defined expressions to expose tangent vectors of the manifold (this is actually a major gripe Hestenes has with the identification of $\partial/\partial x^i$ as a vector, partly because the notion of partial derivatives as vectors is incompatible with the clifford algebra structure).
In this view, then, one can always appeal to the basis vectors of the ambient universal GA and identify a geometrical object as having a fixed expression in terms of the those extrinsic basis vectors. That is to say, clearly a tangent vector (or other geometrically significant object) may be expressed in terms of some coordinate basis vectors intrinsic to the manifold, an expression that will change with different choices of coordinates, but the embedding of the manifold in the UGA isn't changing with the choice of coordinates, so the tangent vector cannot be changing. Intrinsically, one would have to resort to the chain rule, as Anthony did, to identify the change in components as following entirely from function composition (from the chain rule).
Of course, from an extrinsic perspective, it's much easier in general to identify geometrically significant quantities. The covariant derivative of a vector field is just the projection of the derivative of the projection, for instance. Intrinsically, it can be harder to visualize.