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We may suppose that $\overline{s}$ is the algebraic closure of the residue field $\kappa(s)$ of its image $s \in S$. Pick a point $x \in X_{\overline{s}}(\overline{s})$. Set $H = \operatorname{Stab}_{G_{\overline{s}}}(x)$. There is a morphism $f:G_{\overline{s}} / H \to X_{\overline{s}}$. A small argument (valid in any characteristic) shows that in order to prove that $f$ is an isomorphism, it suffices to show that $G_{\overline{s}}(\overline{s})$ acts transitively on $X_{\overline{s}}(\overline{s})$ (here we use the smoothness of $X_{\overline{s}}$). Hence, we are reduced to showing that $\alpha \times \mathrm{id}: G_{\overline{s}} \times X_{\overline{s}} \to X_{\overline{s}} \times X_{\overline{s}}$ is surjective at the level of $\overline{s}$-points. Choose an $\overline{s}$-point $y$ of $X_{\overline{s}} \times X_{\overline{s}}$. It is define over a finite separable extension of the ground field $\kappa(s)$. By smoothness of $X\times X \to S$, we may find an étale $S$-scheme $T \to S$ equipped with a lift $\overline{s} \to T$ and a point $\widetilde{y}: T \to X \times X$ such that $\overline{s} \to T \to X \times X$ is the point $y$ we started with. Now, by assumption the morphism on $T$-points $G(T) \times X(T) \to X(T) \times X(T)$ is surjective, and so there is a $T$-point $ \widetilde{z}: T \to G \times X$ that maps to $\widetilde{y}$. The composition $\overline{s} \to T \xrightarrow{\widetilde{z}} G \times X$ maps to the original $y$, and we conclude the desired surjectivity. This concludes the proof of the claim.

Now we use a hammer. Consider the quotient stack $[X/G] \to S$. By affinness of $X$ and (linear) reductivity of $G$, this has a good moduli space $[X/G] \to M \to S$ in the sense of Alper. The morphism $M \to S$ is flat and of finite type, because the stack was flat and of finite type over $S$. The formation of $M$ commutes with arbitrary base-change on $S$, and it follows from the claim that for all geometric points $\overline{s} \to S$ we have that the base-change $M_{\overline{s}} \to \overline{s}$ is an isomorphism. Hence, $M \to S$ is an isomorphism.

We may suppose that $\overline{s}$ is the algebraic closure of the residue field $\kappa(s)$ of its image $s \in S$. Pick a point $x \in X_{\overline{s}}(\overline{s})$. Set $H = \operatorname{Stab}_{G_{\overline{s}}}(x)$. There is a morphism $f:G_{\overline{s}} / H \to X_{\overline{s}}$. A small argument (valid in any characteristic) shows that in order to prove that $f$ is an isomorphism, it suffices to show that $G_{\overline{s}}(\overline{s})$ acts transitively on $X_{\overline{s}}(\overline{s})$ (here we use the smoothness and separatedness of $X_{\overline{s}}$). Hence, we are reduced to showing that $\alpha \times \mathrm{id}: G_{\overline{s}} \times X_{\overline{s}} \to X_{\overline{s}} \times X_{\overline{s}}$ is surjective at the level of $\overline{s}$-points. Choose an $\overline{s}$-point $y$ of $X_{\overline{s}} \times X_{\overline{s}}$. It is define over a finite separable extension of the ground field $\kappa(s)$. By smoothness of $X\times X \to S$, we may find an étale $S$-scheme $T \to S$ equipped with a lift $\overline{s} \to T$ and a point $\widetilde{y}: T \to X \times X$ such that $\overline{s} \to T \to X \times X$ is the point $y$ we started with. Now, by assumption the morphism on $T$-points $G(T) \times X(T) \to X(T) \times X(T)$ is surjective, and so there is a $T$-point $ \widetilde{z}: T \to G \times X$ that maps to $\widetilde{y}$. The composition $\overline{s} \to T \xrightarrow{\widetilde{z}} G \times X$ maps to the original $y$, and we conclude the desired surjectivity. This concludes the proof of the claim.

Now we use a hammer. Consider the quotient stack $[X/G] \to S$. By affinness of $X$ and (linear) reductivity of $G$, this has a good moduli space $[X/G] \to M \to S$ in the sense of Alper. The morphism $M \to S$ is flat affine and of finite type, because the stack was flat and of finite type over $S$ and $M$ is the relative spectrum of the sheaf of invariant rings. The formation of $M$ commutes with arbitrary base-change on $S$, and it follows from the claim that for all geometric points $\overline{s} \to S$ we have that the base-change $M_{\overline{s}} \to \overline{s}$ is an isomorphism. Hence, $M \to S$ is an isomorphism.

Negative result: If the scheme $S$ lives in positive characteristic, then the answer is negative. For an example, take $S = \mathrm{Spec}(k)$ where $k$ is an algebraically closed field, and take $G = \mathbb{G}_m$ acting on itself $X = \mathbb{G}_m$ via the equation $t \cdot x = t^p x$.

Assume the "torsorness" at the level of the small etale site of $S$. First, we show the following.

$\textbf{Claim:}$ For all geometric points $\overline{s} \to S$, the fiber $X_{\overline{s}}$ is $G_{\overline{s}}$-equivariantly isomorphic to $G_{\overline{s}}/H$ for some algebraic subgroup $H \subset G_{\overline{s}}$.

We may suppose that $\overline{s}$ is the algebraic closure of the residue field $\kappa(s)$ of its image $s \in S$. Pick a point $x \in X_{\overline{s}}(\overline{s})$. Set $H = \mathrm{Stab}_{G_{\overline{s}}}(x)$. There is a morphism $f:G_{\overline{s}} / H \to X_{\overline{s}}$. A small argument (valid in any characteristic) shows that in order to prove that $f$ is an isomorphism, it suffices to show that $G_{\overline{s}}(\overline{s})$ acts transitively on $X_{\overline{s}}(\overline{s})$ (here we use the smoothness of $X_{\overline{s}}$). Hence, we are reduced to showing that $\alpha \times id: G_{\overline{s}} \times X_{\overline{s}} \to X_{\overline{s}} \times X_{\overline{s}}$ is surjective at the level of $\overline{s}$-points. Choose an $\overline{s}$-point $y$ of $X_{\overline{s}} \times X_{\overline{s}}$. It is define over a finite separable extension of the ground field $\kappa(s)$. By smoothness of $X\times X \to S$, we may find an etale $S$-scheme $T \to S$ equipped with a lift $\overline{s} \to T$ and a point $\widetilde{y}: T \to X \times X$ such that $\overline{s} \to T \to X \times X$ is the point $y$ we started with. Now, by assumption the morphism on $T$-points $G(T) \times X(T) \to X(T) \times X(T)$ is surjective, and so there is a $T$-point $ \widetilde{z}: T \to G \times X$ that maps to $\widetilde{y}$. The composition $\overline{s} \to T \xrightarrow{\widetilde{z}} G \times X$ maps to the original $y$, and we conclude the desired surjectivity. This concludes the proof of the claim.

Now we use a double-hammer. Since the stack $[X/G] \to S$ is smooth, the etale local structure of good moduli space morphisms (see Section 10.2 in https://arxiv.org/abs/1912.06162) tells us that, after replacing $S$ with an etale cover, there is a linearly reductive group $H$ over $S$ and an $H$-equivariant vector bundle $E \to S$ such that $[X/G] \cong [V/H]$. By our fiberwise claim, we see that $V$ must be the zero vector bundle. Hence, we upgrade our claim to conclude that, etale-locally on $S$, we have a $G$-equivariant isomorphism $X \cong G/H$ for some linearly reductive closed subgroup scheme $H \subset G$, which is smooth because the characteristic is 0.

Now, if $H$ was not trivial, then we would have some nonidentity section of $H$ over some $S$-scheme $T$ that would stabilize the tautological $S$-section of $X \cong G/H$. Since $H$ is smooth, we may even take $T$ to be etale. This would contradict the assumption on simple transitivity of the action on etale $S$-schemes. We conclude that $H$ is trivial, and hence $X$ is a torsor.

Negative result: If the scheme $S$ lives in positive characteristic, then the answer is negative. For an example, take $S = \operatorname{Spec}(k)$ where $k$ is an algebraically closed field, and take $G = \mathbb{G}_m$ acting on itself $X = \mathbb{G}_m$ via the equation $t \cdot x = t^p x$.

Assume the "torsorness" at the level of the small étale site of $S$. First, we show the following.

Claim: For all geometric points $\overline{s} \to S$, the fiber $X_{\overline{s}}$ is $G_{\overline{s}}$-equivariantly isomorphic to $G_{\overline{s}}/H$ for some algebraic subgroup $H \subset G_{\overline{s}}$.

We may suppose that $\overline{s}$ is the algebraic closure of the residue field $\kappa(s)$ of its image $s \in S$. Pick a point $x \in X_{\overline{s}}(\overline{s})$. Set $H = \operatorname{Stab}_{G_{\overline{s}}}(x)$. There is a morphism $f:G_{\overline{s}} / H \to X_{\overline{s}}$. A small argument (valid in any characteristic) shows that in order to prove that $f$ is an isomorphism, it suffices to show that $G_{\overline{s}}(\overline{s})$ acts transitively on $X_{\overline{s}}(\overline{s})$ (here we use the smoothness of $X_{\overline{s}}$). Hence, we are reduced to showing that $\alpha \times \mathrm{id}: G_{\overline{s}} \times X_{\overline{s}} \to X_{\overline{s}} \times X_{\overline{s}}$ is surjective at the level of $\overline{s}$-points. Choose an $\overline{s}$-point $y$ of $X_{\overline{s}} \times X_{\overline{s}}$. It is define over a finite separable extension of the ground field $\kappa(s)$. By smoothness of $X\times X \to S$, we may find an étale $S$-scheme $T \to S$ equipped with a lift $\overline{s} \to T$ and a point $\widetilde{y}: T \to X \times X$ such that $\overline{s} \to T \to X \times X$ is the point $y$ we started with. Now, by assumption the morphism on $T$-points $G(T) \times X(T) \to X(T) \times X(T)$ is surjective, and so there is a $T$-point $ \widetilde{z}: T \to G \times X$ that maps to $\widetilde{y}$. The composition $\overline{s} \to T \xrightarrow{\widetilde{z}} G \times X$ maps to the original $y$, and we conclude the desired surjectivity. This concludes the proof of the claim.

Now we use a double-hammer. Since the stack $[X/G] \to S$ is smooth, the etale local structure of good moduli space morphisms (see Section 10.2 in Alper, Hall, and Rydh - The étale local structure of algebraic stacks) tells us that, after replacing $S$ with an étale cover, there is a linearly reductive group $H$ over $S$ and an $H$-equivariant vector bundle $E \to S$ such that $[X/G] \cong [V/H]$. By our fiberwise claim, we see that $V$ must be the zero vector bundle. Hence, we upgrade our claim to conclude that, étale-locally on $S$, we have a $G$-equivariant isomorphism $X \cong G/H$ for some linearly reductive closed subgroup scheme $H \subset G$, which is smooth because the characteristic is 0.

Now, if $H$ was not trivial, then we would have some nonidentity section of $H$ over some $S$-scheme $T$ that would stabilize the tautological $S$-section of $X \cong G/H$. Since $H$ is smooth, we may even take $T$ to be étale. This would contradict the assumption of simple transitivity of the action on étale $S$-schemes. We conclude that $H$ is trivial, and hence $X$ is a torsor.

Now we use a double-hammer. Since the stack $[X/G] \to S$ is smooth, the etale local structure of good moduli space morphisms (see Section 10.2 in https://arxiv.org/abs/1912.06162) tells us that, after replacing $S$ with a finite etale cover, there is a linearly reductive group $H$ over $S$ and an $H$-equivariant vector bundle $E \to S$ such that $[X/G] \cong [V/H]$. By our fiberwise claim, we see that $V$ must be the zero vector bundle. Hence, we upgrade our claim to conclude that, etale-locally on $S$, we have a $G$-equivariant isomorphism $X \cong G/H$ for some linearly reductive closed subgroup scheme $H \subset G$, which is smooth because the characteristic is 0.

Now we use a double-hammer. Since the stack $[X/G] \to S$ is smooth, the etale local structure of good moduli space morphisms (see Section 10.2 in https://arxiv.org/abs/1912.06162) tells us that, after replacing $S$ with an etale cover, there is a linearly reductive group $H$ over $S$ and an $H$-equivariant vector bundle $E \to S$ such that $[X/G] \cong [V/H]$. By our fiberwise claim, we see that $V$ must be the zero vector bundle. Hence, we upgrade our claim to conclude that, etale-locally on $S$, we have a $G$-equivariant isomorphism $X \cong G/H$ for some linearly reductive closed subgroup scheme $H \subset G$, which is smooth because the characteristic is 0.