The highest-weight classification does not imply semisimplicity in positive characteristic

Assume the Axiom of Choice inherited from the named suppliers (The Axiom of Choice). The classification of Dominant weights classify the simple rational representations of a split reductive group holds for a split reductive group over every field, but it does not imply that every rational representation is semisimple. Complete reducibility is a characteristic-zero phenomenon (Complete reducibility of rational modules in characteristic zero); in characteristic $p>0$ a split reductive group need not be linearly reductive, as the companion counterexample Rational modules need not be semisimple in characteristic p ↗ on this pair's examples page shows (Milne Example 12.55 and Exercise 12-9; Steinberg Ch. 12, the paragraph after Theorem 39(e)). Readers should not infer semisimplicity of $\mathrm{Rep}(G)$ from the existence and uniqueness of simple modules with prescribed dominant highest weight.

The two statements concern different properties: the classification theorem only asserts that simple modules are parametrized by dominant weights and that the top weight space is one-dimensional, while semisimplicity of every finite-dimensional rational representation is a strictly stronger property that fails already for ${\mathrm{S}\mathrm{L}}_2$ in characteristic $p>0$ (Split reductive groups).