I also have one of these, and mine doesn't quite look like yours. To my eye, your spring doesn't appear to be stretched, but simply too long. Did you or someone else replace the spring since the last time it worked?
Admittedly, my spring as shown above has been over-stretched, but it's what I had, and it provided the proper tension to make things work!
@Chuck Sartor is correct, you clearly need more spring tension than what your video is showing. And as Chuck said, just cut a couple of loops out of the spring (preferably on the screw end) and refasten everything. Just an FYI, if your spring is too long, as yours looks to be, it can end up touching the pulley, thus cancelling any possibility of things working.
The way I adjust mine is to loosen the spring from the solder tab/screw combo, then pull on the loose end of the spring while the power is on, to see just how much tension is needed. You can, at the same time, eyeball where to cut your spring as well. Once you have found the "sweet spot", cut it where there's still enough spring left to rehook it to the solder tab/screw while maintaining the same tension, and make minor adjustments from there.
Also take note of the fact (as shown in my video) that this is a voltage dependent device; the more juice, the faster it goes. I have seen many vibrator motors that won't even move the string except when you are in a very narrow voltage band.
For those devices (and any time I work on any vibrator motor powered device) I recommend taking some 200 or even 400 grit sandpaper (I use 200 wet/dry), fold a small piece in half so that both faces have the rough side out, then stick the creased edge down into the drive pulley (remove the string first!) and gently scuff up the inner groove to allow the string to gain greater traction where it touches the pulley. It doesn't take much, and you'll see the difference right away, and it seems to last a fair amount of time before the string once again "smooths out" the inner fold of the pulley.
One more thing, my string is wrapped around the pulley only one-half turn, and it works great - frankly, I don't know if it should be 1/2 turn or 1-1/2 turns, perhaps someone else can advise?