Well yes, If you think of a quartz watch it generates movement through a small magnet which is given its torque to turn by the magnetic field generated around it, a considerably weak one normally. so often if a quartz watches oils start to dry or a bit of dust gets in the wrong place it grinds to a halt. Introduce a second, much more powerful, higher voltage (yet not powerful enough to overwhelm and damage the wheels) magnetic field generator, one that also oscillates much more rapidly, you can imagine how that will dislodge minor blockages like specs of dust and free up the movement of the gear train etc.
So to answer your question about testing, "kind of" you can't prove a negative (the negative of there being no problem) but if you can find the right spot to place the watch and make the seconds hand do a couple of rotations without the hand hesitating, just progressing smoothly, then it's a good indicator that the gear train and motion work are in reasonable health. Yet that test alone won't tell you if the circuit board or coil are okay, the demag, as far as i know, doesnt work through those parts but directly effects the stepping motor.
Also just a point of clarification in case i wasn't clear, i don't use this jump start method unless the watch fails to start up properly with a new battery.