Gears and rack are worn. Too much play in those areas from your video of the shuttle carrier. With no wear, the shuttle carrier should hardly move when you turn it by hand. You look at the drive pinion, following pinion, short rack and long rack for wear. Also check the bushing that the shuttle carrier shaft goes thru. You might have to replace all of those as well as the bellcrank. I have rebuilt shoepatchers for over 45 years so I know what areas to look at.
glenn