The second part is correct, a glitch buster capacitor is a different role, smoothing the BEC/control power. But I don't suspect that's the solution, at least not yet. The strength of the BEC is an important part of the equation and the Speed Passion GT is reasonably strong, should work at least for the most part.

It's possible the servo has a short, if you can try another servo, or this servo on another RC. Of course simple stuff first, triple check it is plugged in correctly, black/negative wire to the outside of the receiver, no obvious problems with the wire, servo isn't bound up due to being way out of alignment, etc. The light remaining lit is troubling though, usually they at least dim noticeably with these types of issues?

Once working it might need a glitch cap for high load conditions.