So I'm aware that the orbit-stabilizer theorem does not hold for arbitrary spaces with a transitive action by a topological group, but I wonder if it works in the following situation.
Let $G$ be a totally disconnected, locally compact Hausdorff topological group and $X$ a topological space satisfying the same conditions (I would call such things $\ell$-groups and $\ell$-spaces respectively). If $G$ acts transitively on $X$ and $x \in X$ is any point there is an obvious $G$-equivariant continuous bijection $G/G_x \to X$, where $G_x$ denotes the stabilizer of $x$ in $G$. Can we conclude, in this situation, that $G/G_x \to X$ is a homeomorphism? If not, what further conditions do we need to impose? Notice that this is true if $G/G_x$ is compact, since a continuous bijection of compact Hausdorff spaces is a homeomorphism.