I must say that the "carpenter" son seems to be a bit of a non-professional if the description you gave of the installation is correct. Mounting a cantilever arm as described seems to me like it asking for trouble. Even with a smaller sized TV the cantilever arm, when extended, will put a rather large pull out force on the fasteners in the drywall.
To avoid any unfortunate mishap I really suggest getting that short piece of board removed. It should be replaced with a sturdy piece of 0.75 inch thick plywood that spans at least two studs and has a height of at least two feet so that fasteners into the studs can be spaced a good distance apart near the top and bottom of the plywood. Then remount the cantilever arm plate to the plywood. That should give a very reasonable and secure mount for the TV.
The plywood piece can be sanded and painted up nice before attaching it to the wall. If the color matches to the wall it would blend in well and not appear to be out of place behind the TV.
No those anchors will NOT be sufficient to hold up THAT shelf. Those shelf brackets are too short and there will be a LARGE pulling force on the upper screw.
Assuming from the image that's a 10 inch shelf and the screws at 2 inches apart, hanging 79lbs on the edge of the shelf will get you a whopping
395 LBS of pull force on the top screw.
When you look at that picture and look at the metal end, does it remind you of any tool that is sitting on top of your toolbox? A claw-hammer perhaps?
You may get away with the type of anchor that mushrooms out behind the wall as shown below. Remembering to buy the size that matches your drywall thickness. But I do NOT recommend these either for that shelf.
With the shortness of that bracket, I'd still be concerned that was not enough to hold up much weight. If those fail, it is going to pull down a rather large piece of drywall too.
Personally I would be inclined to add a backer board out of a nice piece of wood, the wider the better, using one of the following two methods.
Option 1: Outboard screws... Board cut to fit to next stud width. In your case 3 (Don't forget to add a 2x4 width.). Mount board to studs then centre shelf on the wood.
Option 2.: Inboard Screws. Cut board to about 1 inch wider than the shelf. Then centre the board on the studs as shown in the image below.
Which method is best?
Use the method that gets the screws in the wood closest to the screws in the brackets. However, be aware that with the "inboard screws" method, the end of the board will tend to pull away from the wall if the screws are too far away from the brackets.
If the boards spans more than two studs, I'd add another couple of screws at each stud.
If you, or the Mrs, don't like the visible screws, use one of the many capping or plugging methods.
Best Answer
I would say that the snap toggles are a lousy way to hang a shelving unit like that if only mounting into drywall by itself. If you had any hope of using the shelving unit any where near its load capacity the drywall only solution is a total non-starter.
I see two possible solutions to this situation:
In both cases the wood/plywood can be painted to match the wall color so they do not stand out. Of course in the case of the plywood it could be finished in a way to look like a back for the shelves if it's size matched the full width and height of the shelf unit.