Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some other mathoverflow posts, that any stable sheaf on a K3-surface is simple; however, it is not clear if the underlying field need be algebraically closed. Also, I have not seen a proof of this fact.

The part that I am most confused about is the following. Let $X$ be a (smooth) projective variety over a non-algebraically closed field $k$ and $E$ be a geometrically stable sheaf on $X$. Then, $$\overline{k} \cong \mathrm{End}(E_{\overline{k}}) \cong \mathrm{End}(E) \otimes_k {\overline{k}}. $$ But, Hom-functor commutes with flat base change, in particular with base field extension. This should imply that $\mathrm{End}(E) \cong k$. Is this argument wrong?