I watched and listened to the video and assume that the audio/noise is irrelevant to the question.
I, too, have some movement between the backgear and the pulley assembly when the backgear key is engaged. As with bernard towers' machine this has been the case for many years and so far as I know has never been a problem.
I would think that the only way to reduce this backlash would be to replace one or more components. Your bronze gear appears, perhaps, to be rather worn but this is a different matter.
Edited By ega on 02/01/2022 23:17:47