Proposition: Conservation and trace do not impose Einstein dynamics

Symmetry, conservation, and zero trace of a tensor do not imply that it sources the metric on which it was calculated.

Proof. On three-dimensional Minkowski space, let

It is constant, symmetric, conserved, and traceless for $\eta =\mathrm{diag}(-1,1,1)$. Yet $G_{\mu \nu }(\eta )=0$, while $t_{\mu \nu }\ne 0$. Thus $G_{\mu \nu }(\eta )=\kappa t_{\mu \nu }$ fails for any nonzero $\kappa$.