I believe the answer is "no" for $G$ equal to the split form of $E_8$ over a finite field $k$ of odd characteristic. The set of rational points is a simple finite group. The center of $E_8$ is trivial. For a maximal split torus $T$, I believe the action of the Weyl group $W$ on $T(k)/(T(k))^2$ is trivial, since $W$ is generated by simple reflections arising from copies of $\textbf{SL}_2$ in the group (the "root groups"), and the action of the Weyl group is trivial for $\textbf{SL}_2$. So this would mean that every nontrivial character of $T(k)/(T(k))^2$ gives a counterexample.

Edit. The answer below is incorrect. The correct computation is the determinant of the Cartan matrix, which happens to equal $1$ for $E_8$.

I believe the answer is "no" for $G$ equal to the split form of $E_8$ over a finite field $k$ of odd characteristic. The set of rational points is a simple finite group. The center of $E_8$ is trivial. For a maximal split torus $T$, I believe the action of the Weyl group $W$ on $T(k)/(T(k))^2$ is trivial, since $W$ is generated by simple reflections arising from copies of $\textbf{SL}_2$ in the group (the "root groups"), and the action of the Weyl group is trivial for $\textbf{SL}_2$. So this would mean that every nontrivial character of $T(k)/(T(k))^2$ gives a counterexample.