You're definitely not nitpicking, there shouldn't be any hum/hiss. I could have lived with it as well because as soon as the music played, the hum was inaudible.
I had the same exact issue, but with only 1 out of my 7 4-pin XLR terminated headphones (I didn't have the HD6XX at the time). My issue was a very faint hum in the right side driver of my PS2000e, but what was baffling was that this was the only headphone with the issue, so of course I thought it couldn't be any of my cables or the GSX, so it had to be these particular headphones.
I changed the XLR connector to no avail. I then contacted Grado and sent them in, but they didn't hear the noise (mind you, nobody but myself one other person heard the noise on my GSX and they also heard it with their PS2000e, but not on their own gear) so I figured it was so faint the one other person just doesn't hear it on their gear, nor does Grado. Grado changed out both drivers and the cable, sent them back to me and I had the same humming issue in the right side driver???
I then came across some posts here on this thread about the Mu-Metal and realized I shouldn't have the amp/PSU stacked, so I figured I'd try the shielding even though there was the only headphone giving me the issue, and to my surprise, the Mu-Metal totally solved the issue. And as I mentioned, switching between any gain the amp is dead silent. I still don't have the answer as to why only this 1 headphone had the issue (sensitivity?) but it's resolved and that's good enough for me. Keep us posted.