Drywall – How to mount a TV stand on a wall covered in drywall

Sounds like your sheetrock is somehow damaged from moisture or the EZ Toggle would have certainly worked. You now need to install wood backing, or move your speakers so the brackets can be screwed into the studs.

To install wood backing, first you need to cut out a rectangle of your wall's sheetrock slightly larger than the intended size of the backing panel, with the vertical cuts centered on studs. Don't try to save the old piece, replace it with new. I would use plywood instead of sawn lumber which would make a cold spot. Cut the plywood to fit flush between the studs, preferably with face grain horizontal.

Cut 2 narrow plywood strips to act as cleats. Using construction adhesive, nail/glue the strips to the studs set back from the front edge equal to the plywood thickness. Once the glue has set, nail/glue the plywood backing to the cleats. You are not supposed to nail into the edge of plywood, but in this case it's just to hold the panel in place until the glue sets.

Install a new sheetrock piece to fill the hole you cut out. Tape and mud the joints. Prime paint your finished mud job. If you have textured walls, and don't have a mud gun, there are spray cans of wall texture available for repairs just like this. Either way, you'll need to practice application to match the existing texture. Once dry, apply paint top coat to match existing wall. Whew!

Sure you can't just move the speakers to the studs? ;)

the thickness of a single sheet of drywall

Did they happen to specify how thick that single sheet should be? I bet that they didn't and are being over protective, because they sell drywall in varying thicknesses from 1/8" up to 5/8".

But I don't think they're being TOO over protective, because I think that's the exact same mount I used, and it mounts using lag bolts. Lag bolts are very heavy duty bolts that are driven into the wood and bite into it to create a very strong anchor.

Their strength, however, is in the depth to which they sink in the underlying wood - the deeper they sink the more thread length is biting into the wood holding on tight. So - if you get too shallow of an installation, you don't get the strength you might need.

I read another answer where someone claimed they had a 50" TV mount just in drywall, no studs!

Not for long, I bet...

The other option I considered (although am loathe to attempt) is to cut away the extra drywall, attach some more wood onto the stud (enough to make up why width of the drywall I removed) and attach the mount to that.

You could do that - I thought about doing the same thing - and the way I would do that is to use a multi-tool (below) to cut out the drywall in the same shape as the back plate of the mount, affix that to the studs very securely, mark the stud location on it and drill through the plate into the studs and mount the panel.

HOWEVER - I suspect there's an even simpler solution.

Get longer lags.

They come in a wide assortment of depths and diameters. Just take the lag bolts they supplied with the mount, go to Lowes/Home Depot - and find one that's the same diameter as what they supplied and about 1" longer than what they gave you, and use those instead.