Why in the curved space-time, the double derivatives of the position vector are symmetric but any other vector are not symmetric?

Question

The double derivatives of the position vector (see image eq. (1)), connecting the two points in a curved space-time defined by the Schwarzschild metric, are symmetric under no torsion condition. This symmetry of position vector leads to symmetry of basis vectors leading to the Christoffel symbol symmetry with respect to the lower two indices.

But double derivatives of any other vector are not symmetric (see image eq. (2)) as some of the Riemann curvature tensor components are not zero.

The position vector is a vector, like any other vector and there seems to be no reason why a position vector and any other vector should behave differently in the same space.

Is this because there is no appropriate geometrical framework to describe the curved space described by the Schwarzschild metric?

Kindly refer to the related question:

Is it incorrect to assume the Christoffel symbol symmetry (with respect to the lower indices) for a curved space-time?

Ref: https://physicsoverflow.org/44303/a-classical-scrutiny-of-the-schwarzschild-solution

Answer 1

An answer based the standard interpretation of GR could be:

         "There is no position vector on a manifold. Points can only be labeled by their coordinates and are not vectors. Instead, vectors are elements of the tangent space and the basis vectors are the partial derivatives along the coordinate lines".

This interpretation of the curved space can bypass the questions about geometrical framework of the curved space. But, the vector method gives same results as  the standard tensor analysis, in which the equations are based on the scalar components of the tensors.

The geometrical difficulties and contradictions become apparent when we deal with vectors, as they have both magnitude and directions. 

Ref: 

https://www.researchgate.net/publication/350546500_A_classical_scrutiny_of_the_Schwarzschild_solution