How do we add structurally sound faux stud for a single stud TV mount?
A bunch of ways to do this. What I would probably do myself, which is not what you are currently planning, is:


Cut a piece of plywood, probably 3/4" thick, 24" tall by ~ 20" (studs 16" apart) or ~ 28" (studs 24" apart).
Mount the plywood with 3 screws on each stud.
Paint the plywood to match the wall. It doesn't have to be perfect because it will be mostly hidden by the TV.
Mount the Kanto PS300 to the plywood.


I wouldn't necessarily recommend that for all situations, because of the need to paint the plywood. But if you are going to open up the wall to put in a stud, you are going to need to do drywall patching and then painting, so this will actually be a lot less work.

If you insist on the original plan:


Cut a hole in the drywall ~ 36" tall (24" stud + 4" on top and on bottom for the cross pieces) x ~ 16" or 24" (depending on distance between existing studs)
Screw metal brackets into the existing studs and new 2x4 pieces to span at top and bottom.
Screw metal brackets into the new 2x4 cross pieces and the new stud at top and bottom.
Patch drywall
Paint
Mount Kanto PS300