Combining all the results we see that Riemannian geometry is indeed of a more general nature than axiomatic geometry--at least over real numbers and in dimension strictly greater than 2. One can generalise further by dropping the requirement of a Riemannian metric and retaining the linear or affine connection D. Yet another generalisation is to consider fields other than tensor fields (for example spinor fields) and consider derivations on such fields.
At the same time a number of interesting questions remain if the field is not that of real numbers (in dimension at least 3) or if we have a planar geometry which does not satisfy Desargues axiom (which is almost obvious in dimension 3)--in the latter case we do not even have a skew-field associated with the geometry. These are the subject of study in Combinatorial Geometry and Group theory. The latter (group theory) points to another approach to geometry--that taken by Pieri, Klein and Lie--where the fundamental notion is that of a group of motions. A similar study to the one undertaken above ought to show that this leads to Lie groups and Symmetric spaces.