Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics?

The answer is no. This relies on two things:

  1. A uniquely geodesic proper metric space is contractible; see here for a proof.

  2. $C_n$ is not contractible. Indeed, it has many nontrivial homology groups (there is a huge literature on this).

This is just a long comment, and a pretty speculative one at that. However it might perhaps be of interest to you since:

  • there is a natural connection to physics,
  • the construction only works in three dimensions,
  • the construction is equivariant with respect to the action of the permutations.

I have in mind the (conjectured) map described by Atiyah in [1] which maps configurations of points to the complex flag manifold: $$ C_n(\mathbb{R}^3) \to U(n) / T^n. $$

Since the flag manifold is homogeneous, this map would provide a metric on $C_n(\mathbb{R}^3)$ if we could spot a natural metric on the fibres. I don't know if this is possible but for $n=2$, the fibres of the map are pairs of distinct points defining the same direction (first point looking at second point) and so are naturally parameterised by their midpoint $m$ and distance apart $t$. It's a bit of a stretch but if we give this fibre the metric of $dm^2 + (dt/t)^2$ then we get something you might regard as "nice".

[1] Atiyah, M., "Configurations of Points", R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), no. 1784, 1375-1387.