Most of the drywall wisdom in the audio world pertains to soundproofing. Very few audiophiles choose in-wall speakers, even though theoretically they are superior because they eliminate the main source of acoustic problems, echos from the front wall. It is a myth that in-walls can't provide the illusion of depth.
Insulating inside the wall with FG is the best practical thing you can do. The biggest problem is the ringing of the drywall on the studs at upper bass and midrange freqs. It's like a drumhead on a rim. Drywall resonance sounds terrible, so quieting it will give you more midrange refinement. Insulation will damp air motion inside the bays, which will damp drywall motion.
Making the wall stiffer is a secondary concern to filling the voids with FG, and more difficult to do. But stiffening the wall will tighten bass transients, make more low bass slam and reduce smearing the mid bass as the wall acts like a giant passive radiator. But stiffening an existing wall is difficult. If you had nothing better to do with your time and money you could rebuild the wall from scratch with 6" steel studs, 8" on center to make a truly stiff wall. Or you could hire a foam insulation contractor to spray urethane foam into the existing stud bays, thus gluing the front and rear drywall "membranes" together and creating a foam sandwich structure that is very stiff, and that would quiet the ringing drywall too. They do this on existing construction to replace old paper insulation. Or even just make a brace of some kind on the back of the wall, maybe a 2x4 attached diagonal from behind the speakers to the side walls.