$\DeclareMathOperator\SO{SO}$Fix even $n=2m\ge 4$. Let $G_0$ be the copy of $\SO(m)\times\SO(m)$ in $\SO(2m)$. Let $G$ be generated by $G_0$ and an element of order 2 switching the two copies of $\mathbf{R}^m$. Let $x$ be a vector whose decomposition $x_1+x_2$ according to the orthogonal decomposition $\mathbf{R}^m\oplus\mathbf{R}^m$ satisfies $0<\|x_1\|<\|x_2\|$. Then $G$ satisfies all conditions: each orbit has dimension $\ge m-1$, $\phi$ (the homomorphism with kernel $G_0$) is surjective, and $x$ has stabilizer contained in $G_0=\operatorname{ker}\phi$.