(Edit: the previous version of this answer was not correct, but I leave this here as a remark)

Let $\sigma: G \rightarrow G$ be the Frobenius map corresponding to the field automorphism $x \mapsto x^q$.

To show that there exists a regular semisimple element in $G_{\sigma} = G(q)$, by the Lang-Steinberg theorem it would be enough to show that there is a $\sigma$-invariant class of regular semisimple elements. See 2.7(a) in "[Conjugacy Classes](https://doi.org/10.1007/BFb0081546)" by Springer and Steinberg (in  [Lecture Notes in Math 131](https://doi.org/10.1007/BFb0081541)).