Fester,
Looking at your video, that lower pin shouldn't rock like that. If both of the springs (the bent wire and the flat spring steel ones that interconnect to support the pins) are correctly mounted and pressing out on the upper and lower pins, they shouldn't rock like that.
It's possible the handle is splayed out some and/or the holes might be wallered out. You may also want to check and make sure the "bushing" (Parts Group 4.016) in the exploded parts diagram above is intact and not worn out as that is supposed to help retain the base of the curved wire spring on the inner shaft arm.
If it does turn out to be the handle, there is a supposedly NOS GM# 3716517 (not mine) being offered
HERE . The paint on it is chipped up, but looks unmolested otherwise. A decent asking price (not gold plated - haha).
Let us know how things turn out.
Dan
