The stable locus has a geometric quotient

The subset $Y=\{y:\dim G_y\ge 1\}$ is closed in $X$ and $G$-stable by [F3].

Let $x\in X^s$. Then $\pi (x)\mathrm{\notin }\pi (Y)$: if $\pi (y)=\pi (x)$ for some $y\in Y$, then the fibre $F={\pi }^{-1}(\pi (x))$ is closed, $G$-stable and contains both the closed orbit $Gx$ and $Gy$; by the unique closed orbit property [F2] the orbit $Gx$ is the unique closed orbit in $F$, so the closed orbit contained in $\overline{Gy}$ by [F5] must be $Gx$, giving $Gx\subseteq \overline{Gy}$. Since $\dim Gy=\dim G-\dim G_y\le \dim G-1<\dim G=\dim Gx$, the point $x$ is not in $Gy$, so $Gx$ lies in the boundary of $\overline{Gy}$; but every orbit in that boundary has dimension strictly smaller than $\dim Gy$, by [F5], contradicting $\dim Gx=\dim G>\dim Gy$. Hence $\pi (x)\mathrm{\notin }\pi (Y)$.

By [F4] applied to the closed $G$-stable set $Y$ and the point $x\in X^s$ of step 2.1, there is $f\in \mathbb{C}[X{]}^G$ with $f(x)\mathrm{\ne }0$ and $f{\mathrm{\mid }}_Y=0$. Put $X_f=\{f\mathrm{\ne }0\}$. Since $f$ is invariant, $X_f={\pi }^{-1}(D(f))$, so it is open, saturated and $G$-stable. Every $y\in X_f$ satisfies $y\mathrm{\notin }Y$, hence $\dim G_y=0$ and $\dim Gy=\dim G$. If $Gy$ were not closed, its boundary would contain an orbit of dimension strictly smaller than $\dim Gy$ by [F5]. But $\overline{Gy}\subseteq {\pi }^{-1}(\pi (y))\subseteq X_f$, since the quotient fibre is closed and contains $Gy$; every point of this closure lies outside $Y$, so each orbit in the boundary has dimension $\dim G$, a contradiction. Thus $X_f\subseteq X^s$. Applying the same construction to every stable point gives $X^s={\bigcup }_f{X}_f$, where $f$ ranges over invariants vanishing on $Y$.

Consequences for openness and saturation: $X_f={\pi }^{-1}(D(f))$, so $\pi (X_f)=D(f)$ by surjectivity of $\pi$, so it is open in $X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ and $\pi (X^s)={\bigcup }_{f}D(f)$ is open. If $\pi (x^{\mathrm{\prime }})=\pi (x)$ with $x\in X^s$, choose the invariant $f$ constructed at $x$ in step 3.1; then $\pi (x)\in D(f)$, so $x^{\mathrm{\prime }}\in {\pi }^{-1}(D(f))=X_f\subseteq X^s$. Thus $X^s={\pi }^{-1}(\pi (X^s))$ is saturated, and it is open and $G$-stable in $X$.

Structure sheaf: let $f\in \mathbb{C}[X{]}^G$ with $X_f\subseteq X^s$. By [F8] the invariant regular functions on ${\pi }^{-1}(D(f))=X_f$ form $(A_f{)}^G$ with $A=\mathbb{C}[X]$, and the Reynolds localisation identity [F7] identifies this with $(A^G{)}_f={\mathcal{O}}_{\pi (X^s)}(D(f))$, so the pullback along ${\pi }^s$ is an isomorphism onto the invariant functions on each principal piece; the identity is compatible with restriction and glues over the basis of the $D(f)$. This is condition (iii) of [F6], and it also gives ${\mathcal{O}}_{\pi (X^s)}=({\pi }_{\ast }^s{\mathcal{O}}_{X^s}{)}^G$.

The fibres of ${\pi }^s$ are exactly the orbits: if $\pi (x^{\mathrm{\prime }})=\pi (x)$ with $x,x^{\mathrm{\prime }}\in X^s$, then $Gx$ and $G{x}^{\mathrm{\prime }}$ are both closed orbits in the same fibre, so they coincide by uniqueness of the closed orbit in that fibre [F2]. Together with surjectivity of ${\pi }^s$ onto $\pi (X^s)$ this gives condition (i) of the geometric quotient [F6].

Quotient topology: let $W\subseteq X^s$ be open. Then $W$ is open in $X$ because $X^s$ is open, and $G\cdot W$ is open, $G$-stable and satisfies ${\pi }^{-1}(\pi (G\cdot W))\cap X^s=G\cdot W\cap X^s$ by step 5.1. The complement $X\setminus G\cdot W$ is closed and $G$-stable, so its image is closed in $X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ by [F2]; intersecting with $\pi (X^s)$ gives $\pi (G\cdot W)\cap \pi (X^s)=\pi (X^s)\setminus \pi (X\setminus G\cdot W)$, which is open in $\pi (X^s)$. Since $\pi (W)=\pi (G\cdot W)$, condition (ii) of [F6] holds: a subset of $\pi (X^s)$ is open exactly when its preimage in $X^s$ is open.

Conclusion: by steps 4.1 and 5.1 the map ${\pi }^s:X^s\to \pi (X^s)$ is surjective with fibres exactly the orbits, and by steps 6.1 and 4.2 it satisfies the quotient topology and invariant-function conditions, so it is a geometric quotient by [F6]; the orbits in $X^s$ are closed in $X$ by the definition of stability [F1], and the sheaf identity of step 4.2 completes the statement. All Axiom of Choice content is inherited from the quotient, semicontinuity, separation and orbit suppliers used above.