An item like that to spring in to existence without any record of a history leading up to it or a history of devices descending from it really makes you wonder whose imagination it sprang from and why it wasn't recognized that these principles had further application, or, alternatively could point to lots of stuff (including knowledge) getting lost.
Where we might be today if that had not happened...
An explanation for that fact that so few devices remain that a record of history seems to be missing, could be that they were made of metals (copper, bronze, etc) which were quite precious in that time, so the devices probably got recycled. Image that, some warrior smelting a few antikythera machines to forge a sword!
I think you got confused by the wording of "precision of the original metalworking and gearing", original as in 'not the lego version', not suggesting it was the first use of gears.
Someone correct me if I'm wrong, but it seems like what is important here is simply the ratios of the number of teeth on the different gears.
I mean obviously all the parts have to mesh when the device is cranked but it doesn't have to turn terribly smoothly to work.