This whole discussion looks sooo familiar. I have had limited success adjusting the black plastic bottom bearing piece, and like others mentioned, I moved it around with the motor running, while held in my hand. Once I found the best spot, I shimmed and CA glued it in place. The other trick I've had some success with is loosening the brush plate just enough to allow it to move slightly, but not enough to provide any vertical movement. Then with motor in hand and power applied, twist the brush plate one direction, then the other and I'm fairly certain you will see a slight difference in both ease of startup, and a higher rpm for a given voltage. Once you find that sweet spot, tighten the screws. That is something I now do every single time when I remove the brush plate.
I have found this type of motor with the aluminum side plates and the plastic bearing plate, to be very touchy as far as making it run well. Even on the best of them, once adjusted for near perfect performance, as things warm up within a few minutes of running, I can detect differences in the ease with which it starts. One of my better runners when cold, will actually stall when hot and trying to pull a moderate load. When it cools back down, back to smooth, perfect starts again.
The same goes for your style with no ball bearing in the plastic bottom plate, but instead has a bushing mounted in the aluminum chassis under the armature - that style seems to develop excessive side play, which I would guess is due to greater forces applied to the bushing by the worm gear, which is caused by more leverage in the whole motor assembly arrangement.
I understand there is an alignment jig available to help align the brush plate, the field stack and armature, and the bottom bearing plate, but I have not seen one and have no clue how well they work. The older motors like in the 622, 6250, etc., are not made this way and do not seem to have this problem.
And yes, the pinion gears and their little shafts can absolutely be a factor in binding, but they really do have to get pretty bad before things actually bind up - from your video, yours did not appear to be that bad.