The craftsman in the film uses a fairly wide band saw blade -- 3/8 inch perhaps. Despite its large sized, he shows how to used a series of straight cuts and the front of the blade to nibble away and tight and curved area.
He does turn to the scroll saw, not for cutting, but for filing the edges of the pinion smooth. He mounts a needle file into the scroll saw. You can also buy files designed to fit in the scroll saw.
The video also shows how to use a benchtop disk sander to create round spacers and how to best glue the parts together with white PVA glue.
This is the kind of education you can only get by watching someone with experience.