Are resource savings for the evaluation of $k$ dependent instances in parallel possible?
Assume $C'(X)$ is a logically equivalent but minimized version of $C(X)$. Then $C'(X) \notin  SIZE(poly(X))$, unless the polynomial hierarchy collapses. 

To see this, assume $C'(X) \in SIZE(poly(X))$, i.e. $C'(X) \in P/poly$. As noted in the original post, this would imply $NP \in P/poly$, but then the claim follows from Karp-Lipton.

Thus, even the existence of a poly-sized circuit equvialent to $C(X)$ is unlikely. Or, in turn, a generalization of direct product theorems to the dependent case appears plausible under complexity theoretic assumptions.