Assume that $G\leq\operatorname{Sym}(X)$ is a permutation group generated by all its point stabilisers, i.e. $G=\langle G_x \mid x\in X\rangle$. There is no cardinality restriction on $X$. Furthermore, assume that $G$ has *finitely many* orbits, and that $G$ is subdegree-finite, i.e. all point stabilisers have only *finite* orbits.

Is $G$ then necessarily generated by *finitely many* point stabilisers?

The question arises in the context of the permutation topology. When we endow $G$ with the permutation topology, $G$ is a totally disconnected locally compact group when all $G_x$ are compact, and I want to know if having finitely many orbits is then sufficient for $G$ to be compactly generated.