Proposition: Tensor identities alone do not imply Einstein dynamics

Symmetry, conservation, and zero trace of a tensor do not imply

Proof. On four-dimensional Minkowski space, choose a nonzero constant null covector $k$ and put $t_{\mu \nu }=k_{\mu }{k}_{\nu }$. This tensor is symmetric and nonzero. It is conserved because it is constant and traceless because $k$ is null. The Minkowski Einstein tensor vanishes, so the displayed equation fails for $G_{\mathrm{N}}\ne 0$.