A stronger result of the theorem, Action is primitive iff $G_x$ is maximal for all $x \in X$

Question

I'm reading through Stephen Lovett's Abstract Algebra on group actions.

I understood the proof of Proposition 8.3.8, that if $G$ acts transitively on $X$, then the action is primitive iff $G_x$ is maximal for all $x \in X$.

But then, using the fact that $G_{gx}=gG_xg^{-1}$, Lovett claims that the Corollary is a stronger result:

I do not understand why this is a stronger result. I also do not understand parts of the proof, such as equation (8.5), and why it seems as though only one direction of the proof is given. Any help would be much appreciated. Thanks!