XLR pin 1 would usually be connected to..
Ah, "ground", JR, truer words are rarely spoken.
There are more than a couple of flavors of ground, and it can definitely help to keep track of who's who and when and where they want to play together, or not. The XLR pin 1, as well as the TRS sleeve, connection is not signal ground. It therefore should not be lumped in with the PS and signal grounds inside the gear. Pin 1 is the shield drain. If you attach it to your signal ground on the PC board in your gear, any current that the shield is trying to shunt will transfer to said ground. Shield should connect to shield, and the chassis is the gear's shied. Pin 1 should connect to the chassis directly. Do not pass Go, do not collect $200. In fact, there are many pieces of professional gear who's noise performance can be markedly improved by severing the PC mount XLR pin 1 connection to the ground plane and wiring them straight to the chassis. Don't ask how I know.
Neil Muncy's paper (preprint 3930) presented at the 97th AES convention (AES Journal 6/95) "Noise Susceptibility in Analog + Digital Signal Processing Systems" is one of the definitive bits on this. In it, he defines the "Pin 1 problem". This is what one gets by attaching pin 1 to the ground plane, or signal ground, like the aforementioned manufacturers. Via common impedance coupling, all sorts of nasties can show up in the audio. Kind of like daisy chaining power strips together or shorting the neutral and ground in an iso ground AC run.
I read a more recent interview with him where he mentions that the manufacturers have gotten hip to it, and the problem is becoming less prevalent. I just have to say, he must only work with nice toys. So many mid level and "prosumer" bits of gear are done so far on the cheap, they wouldn't connect anything at all if they didn't have to, much less if doing it right cost 3 cents.
Anyway, that paper, and Kenneth Fause's "Fundamentals of Grounding, Shielding and Interconnection" from the same convention will school one proper on this sort of thing. I can't find them online, though I'm sure you could buy them from AES should the fancy take you. Quite a few good treatises footnote one or the other, though. A search on "pin 1 problem" should provide a good bit of reading.
I want to mute each input (including it's ground)
Balanced signals don't really have a ground, per se. (Somebody's going to kill me for that, I'm sure). The signal is the difference between the "hot" and "cold" legs, rather than the difference between "hot" and "ground", "cold" and "ground", or some combination, so I wouldn't worry too much about switching the shield out of circuit unless you're having other interference troubles.
Anyway, long story short, (I know, too late.) pin 1 to chassis, and it's not out of the question to lift one side in the cable. The only XLRs that absolutely need shield connected through are ones used to phantom power microphones.