[hansen]

That gives you only the equation of motions for a test particle, i.e. the geodesic equation. The field equations are another thing. I think historically there was a lot of guess work. But using that the sources of the field are the energy/momentum density you can make them plausible: looking for a divergence free 2-tensor that depends on the curvature the Einstein tensor is easiest choice. The coupling constant can be determined by looking for the non-relativistic limit which must be identical to Newtons law.

On the other hand, the Einstein-Hilbert action[1] is so beautiful it just has to be right :) It simply says that the the solutions to the field equations are the stationary points of the averaged scalar curvature.

[1] https://en.wikipedia.org/wiki/Einstein-Hilbert_action