With the cock removed (as in your video) there will be a lot of play as the jewel that supports that end of the pinion is in the cock.
Also, I can't see any sign of the tension spring in any if the images. The specific function of that spring is to control seconds hand stutter so without it there will be a problem. You mention that without the cock the spring sits up in the air above the pinion, which suggests that the spring isn't actually missing, but it also suggests that the spring should sit underneath the pinion leaves so that it lifts the pinion up into the jewelled bearing in the cock thus providing the tension to control the seconds hand. I have not worked on one of these so I don't know if the spring should lift the pinion up or press it down, but either way it needs to be in contact with the pinion.