Showers/tubs should have a access panel on the other side of the wall the faucet is on so you can get at the plumbing. Mine didn’t (also 1960’s house) so I cut an access hole and got one of those spring loaded wall hole covers that are designed to allow wall access without cutting every time.
https://www.homedepot.com/p/Owens-Corning-R13-Thermafiber-Fire-and-Sound-Guard-Plus-Mineral-Wool-Insulation-Batt-15-in-x-47-in-1310109/319880476
This is the kind of thing I would suggest, it’s designed to be soundproofing, fire retardant home insulation. This is literally the exact thing you would put in-between your joists before drywall if you were trying to insulate an interior wall for sound.
There’s a product called green fiber that is a blown-in cellulose insulation that’s been treated with fire retardant you could use if the walls have already been drywalled, I’ve used that in all of the interior walls of my home when I was remodeling to try and deaden sound and have added privacy throughout the home. My basement, which I turned into a mother-in-law apartment, is so private I can’t hear my basement neighbor when they have their video games blasting through a surround sound stereo set with a subwoofer. I can feel the floor shake but, unless I am standing right next to a vent, I can’t hear anything.