In the classic book by Misner, Wheeler and Thorne, they justify the form of the Einstein tensor, $G$, by the fact that it is the unique tensor which satisfies
- $G$ vanishes when spacetime is flat
- $G$ is a function of the Riemann curvature tensor and metric only
- $G$ is linear in the Riemann curvature tensor
- $G$ is a symmetric and a second-rank tensor
- $G$ has vanishing divergence.
Point 1 comes from the fact that if we equate gravitation to geodesic deviation by spacetime curvature, then the absence of spacetime curvature should mean no gravitation. Point 2 basically says gravitation is due to geodesic deviation only. Point 4 is basically because curvature is a two-form (or since we want the stress-energy tensor to be a source of spacetime curvature, which is a rank-two tensor) and point 5 is due to (local) energy-momentum conservation.
I do not understand why we need point 3 though. Possibly at the quantum level there might be corrections which are non-linear in Riemann curvature tensor, but why, at the classical level, do we demand linearity in the Riemann curvature tensor?