The residual plateau algebra is trivial

This does not say that all residual parameters on Minkowski space vanish. Smooth noncompact wave solutions exist. It says that the frozen plateau support class contains none except zero.

Consequently ${\mathfrak{g}}_U$ is not a nontrivial residual action on classical de Donder free solutions. More generally, a compact shift $K\xi$ preserving the gauge-fixed free equation would require $K\square \xi =0$; the compactly supported vector $\square \xi$ would then be a Killing vector and hence zero, again giving $\xi =0$.

The admitted plateau algebra remains an algebra acting on the classical background prescription. It is not thereby promoted to an interacting action on the free graviton algebra. The graded transformation involving the ghost is a different construction. In particular, ghosts are not required to be compactly supported plateau wave solutions.