Actually, I can answer my own question: many cliques of size $4$ which are separate to one another. The graph is not $3$-colorable, but the induced subgraph is empty.

Yes, I see what's the problem is. Would you say that its correct to assume that $G$ is $3$-colorable iff the induced subgraph among all vertices with degree at least $\sqrt{\left| V\right|}$ can be colored as described by Wigderson's algorithm?

Do you mean that bounding the out-degree by a constant is hard? For my question, also if the bound is small, but depends on a parameter of the graph it might be ok (Such as $2\cdot a(G)$, only its not small enough, its hard for me to define what is considered small enough for me, but I'd like to see several possibilities, if they exist).

@D.W. I mean that you have single bit inputs $x_1,...,x_n$, and $s$ is telling me for each $i$ whether $x_i =1$ or $x_i=0$, and in $CVAL$ Im checking whether the output of the entire circuit is $1$ or $0$.

@YuvalFilmus I know, but I'm trying to prove here that it is also $RP$-hard (or perhaps complete), through similar way, but not sure it can be done so.