Suppose that we take principle of least action as given. Also assume that any manifold allowed by the action would carry Levi-Civita connection (torsion-free characteristic). Also assume that the local symmetry imposed on the tangent space of each manifold point is that of Poincare group, via general covariance principle.
Would these be sufficient to derive Einstein-Hilbert action, and by corollary Einstein field equations? Or do we need extra conditions to derive the Einstein-Hilbert action?
Edit: If not, then what would be other extra conditions?