Acceptable joist depth for a second set of ceiling-only joists
You’re dealing with “impact sound control “ not “airborne sound control “.  Stacking 2 layers of 5/8” gypsum board on “hat channels “ is not the best way to control impact sound.
First
We control impact sound (walking in high heels, dropping objects, etc.) at the source by using

soft materials to walk on, (carpet, cushioned vinyl flooring, etc.), and
isolate the underlayment from the subfloor and joists with sound absorbing fiber board. If you don’t have access to the floor above your unit, then adding sound control below is difficult...but not impossible.

Second
Isolating your ceiling from the floor above is correct, but using Resilient Channels, not hat channels. The transfer of impact noises down to your space needs to be isolated....that’s why adding ceiling material to the bottom of the floor joists does not work AND WHY using hat channels does not work as well as Resilient Channels.
Yes, the new ceiling joists (isolated from the floor joist above) helps isolate the ceiling from the floor above, but impact noises travels through the framing. Hat channels do not isolate the impact noise.

By the way, your loading seems a little light: 

two layers of 5/8” gypsum board is 6.25 psf,
hat channels is about .5 psf,
new 2x6 joists at 16” oc is 1.5 psf,

PLUS, I would add:

rock wool insulation at 4” at 2 psf,
1/2” fiber sound board at 2 psf and misc. (lights, etc.) at 1 psf...

for a total dead load of 13.25 psf. (Even the crappiest grade of 2x6 at 16” oc will support that for 12’)
I’d recommend:

isolating the ceiling joists from the wall by installing sound board between the new ceiling joists and the wall,
installing 4” of rock wool in the existing floor joists, in the stud spaces and on top of the new ceiling,
don’t install your light in the ceiling (cutting s hole for a wire allows sound into your space,
install 1/2” sound deadening board on top of your new ceiling joists (if possible,
install 5/8” gypsum board on RC -1 channels on gypsum board on RC-1 channels on the 2x6 ceiling joists,
caulk the perimeter with sound control caulk...leave the 1/4” spacing at the wall as per your insert you referenced.

Yes, two layers of gypsum board is better than one, and separating the layers with RC channels is best.