Good lord.
Well, if it is an HD (and it seems to be), then the unit as a whole has a serious defect, and should be returned.

But then, one asks why.
One possibility is that the flexspline, which is the heart of the unit, is really seriously worn out. Just how this could happen, I do not know, and if it did happen I would expect the whole thing to collapse.
Another possibility is that the outer part of the wave generator has been worn out and is far smaller than it should be. Same comments.

Another possibility is that something is actually missing or is of the wrong size. This seems like a possibility, as the bit on the stepper motor did not look as though it had any bearings. I note that you are able to push the motor drive back in place fairly easily: this should not be possible they way you did it, except with great care and skill. I would prefer to say it looks wrong. And when you wobbled the front end, I could see the flexspine wobbling back and forth.

Underlying all this is the question of how could the thing backlash like that. It could be that the spline is not fully engaging with the outer teeth, but is instead able to rattle from side to side - like rattling poorly engaged gear teeth in a conventional gearbox. This would be consistent with a missing or wrong-sized part.

Have a look at the attached picture (from HD catalog). Does the drive bit look like this, with either balls or rods as bearing elements? Some detailed close-up still pics would be more useful now than videos.

Incidentally, that unit is a 14-50 one according to the label on top: the smallest in the whole range. It has a rather large chuck mounted on it. In my book, the chuck is a bit big for the bearing.

Cheers
Roger