Why does it suffice to show that $V$ is constant on each sphere if we are trying to show that $F(x) = - \text{grad} V(x)$ and $V(x) = g(|x|)$ implies that F is central?
Wouldn't a constant $V$ simply mean that $F$ is 0? That would satisfy the condition for being central ($F(x) = \lambda(x) x)$ if $\lambda = 0$, but what about when $\lambda \neq 0$?
If the level sets of $V$ are spheres centered at the origin, then $\text{grad }V(x)$ is normal to the sphere at $x$, hence a scalar multiple of $x$. (The equation $V(x)=g(|x|)$ says, of course, that $V$ is constant on spheres $|x|=\text{constant}$.)