The watches have different first-person and third-person offsets (position from the camera and hand) The watches are invisible in third person because the third person offset is thousands of metres below the ground. The watches used the state system for some controls (charging ticks, readying) which unmounting the image would stop, although that'll work for different cases.
You can see the first/third person effect in some items like the Print Gun and Wrench - in first person, they don't move at all when walking or change position when fired, as compared to the Gun which bobs up and down and shifts when firing, moving bricks, jumping.