I've had my alternator pulley seize up so it was just solid. The bouncing tensioner was much worse than in your video. So yours is probably okay.
It's easy to check that the alternator pulley is working properly. Just take the serpentine belt off, stick a screwdriver or something into the cooling fan blades inside the alternator so you can stop it from turning, and then try to turn the pulley by hand. It should freewheel in one direction and not the other. (Do this with the engine off, obviously!)
When mine failed it went solid in both directions, so with the alternator blocked from spinning, the pulley would not spin in either direction.