Why?

Many modern watch parts are CNC machined, often the finishing is done by hand such as zaratsu polishing - but even that is a repetitive motion that can be mechanised.

I would not be surprised if given enough time even what we have today - a decent VLA model + some other specialised models, 6-axis CNC machine, an SMD pick and place etc would be capable of designing, manufacturing and assembling a mechanical watch.