I might well be missing something here, but:
Consider $S = \mathbb{P}^2$ and $E$ the tangent bundle to $\mathbb{P}^2$. Set $G = GL(E)$. If $T$ is a maximal torus of $G$ then,for every point $x \in \mathbb{P}^2$, we have a maximal torus $T_x$. The eigenspaces of $T_x$ form two points in $\mathbb{P}(E)$; let $\Lambda \subset \mathbb{P}(E)$ be the set of eigenspaces of the maximal torii. Then $\Lambda \to \mathbb{P}^2$ is a double cover. Since $\mathbb{P}^2$ is simply connected, $\Lambda$ has two connected components. Let $L_1$ and $L_2$ be the sub-line bundles of $E$ spanned by these components. Then $E = L_1 \oplus L_2$. But a standard computation with Chern classes shows that $E$ is not the direct sum of two line bundles.
Of course, this example is neither affine nor has vanishing Pic, as you last paragraph requests.