How to soundproof a wall between two apartments in a 1950s NYC structure?
You don't need both quiet rock and green glue, they should be seen as alternatives.
Also you're going for higher frequency noise: footfall or low frequency noise would be a different answer.

I'd do this: identify and mark stud locations, have a contractor drill holes and dense pack insulation into the wall (see video), patch the holes, find your marks then green glue and screw 5/8" drywall and be done with it.  No framing needed.

But your next problem is likely flanking: sound coming through the floor, heating ducts, over the top plate of the wall, out and back in a window, etc.  You'll likely still hear that TV without some pretty detailed work to find all the paths.