Case 1: Bridge Air Gap By Fastening to Building Structure With Overhung-load Bearing Fasteners
The air gap problem is quite common: The sheetrock on many walls I have encountered doesn't contact the studs everywhere. So, when you tighten down a TV mount to the wall, the drywall deflects toward the studs. You can easily cause cracking of the drywall, particularly when there is a seam in the area.
The principle of the solutions in this case make use of a fastener's ability to support an overhung load.
The fastener and the building structure have to be strong enough to withstand the torque created by the downward force of TV and Mount acting over the distance between the drywall surface and the building structure.
Case 1.1: Use Hanger Bolts
Assuming you find wood building structure in there (furring doesn't count here because it will twist under the torque), I recommend hanger bolts.
Use a spade bit to clear a hole in the drywall big enough to accommodate a nut flush with the surface of the drywall. Then put the mount over the hanger bolt and tighten another nut down on that. The first nut will keep the drywall from deflecting toward the wall. Size the hanger bolt larger than normal because of the overhung load. From your description I would guess that 4x 1/4" hanger bolts would be adequate.
Make sure you pilot drill for your hanger bolt or you risk splitting whatever wooden structure you're screwing into.
Case 1.2: Variation for Concrete Construction
I've also encountered furred out concrete clad in drywall (common in highrise condominiums). In that case, the same principal as Case 1.1 applies, but use "drop-in anchors" and a threaded stud instead of hanger bolts.
Case 2: Fasten to Furring with No Air Gap
Sometimes drywall is attached to and is in contact with (i.e. no air gap) furring.
Case 2.1 Furring Attached to Building Structure
Where furring is attached directly to the building structure, drive multiple wood screws directly into the furring. If there is some hard material behind the furring, ensure that the screws are short enough. Otherwise they can jack the furring from the building structure when their points impact cinderblock or concrete.
The principle of this solution is to transfer the load to the building's structure directly through the furring.
Case 2.2 Furring Dubiously Attached to Building Structure
Where furring and drywall are creating their own structure, drive multiple wood screws into the furring to fasten your TV mount.
The principle of this solution is to use the furring to distribute the load to a large area of the composite structure of furring and drywall.
Drywall itself is rather heavy, so if the wall is able to hold itself up, there's a good chance it can hold up your TV to provided you distribute the load over a large enough area of drywall and furring.
Another way:
Vesa MIS-C mount on the TV?
Presumably the entertainment center shelf is designed to take considerable weight.
You could run a big L bracket down to the floor of the shelf, where you've installed a slot for the bracket to slide into and secure the display.
Since 1" screws are unlikely to hold the full leveraged weight of the display, you could pile 50 or more pounds of weight onto the top of the slot structure. That should eliminate the possibility of your screws yanking out of the cabinet, and also eliminate your having to put screw holes in the, no doubt pretty, front of the cabinet.
Best Answer
Assuming the cross bar only has two mounts you only need the two mounts to come out 1/2" from where they would normally site.
I'd just cut a couple squares of 1/2" plywood and then either screw the squares to the studs and the mounts to the squares or if the weight of the cross bar with backdrop is significant then get screws that are 1/2" longer than the ones you would have normally mounted with. The squares then just act as spacers to push the cross bar out the 1/2" front the face of the wall.