[Arran]

 In answer to your question, yes I do think that piece is supposed to move up and down now that I have seen a better view of the mechanism in the video. The motor shaft is stepped, and for the platter to change speeds the idler wheel is supposed to move up and down when the speed selection lever/knob is moved. I would try one of the old standbys like WD40 or PB blaster to see if you can free it up, then take it apart and clean whatever old grease is left on it in mineral spirits, relube, then reassemble. Hopefully the cast zinc itself hasn't deformed or something, otherwise you will be looking for a parts unit or some machining work.
Regards
Arran