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

The set of regular semisimple elements is invariant under $\sigma$. (If $g \in G$ is semisimple, then $g$ is regular semisimple if and only if $C_G(g)^\circ$ is a torus.) 

Thus it follows from 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)) that there exists a regular semisimple element fixed by $\sigma$. 

The proof of 2.7(a) does not need machinery other than the Lang–Steinberg theorem.