Why does the 손 없이 bind to the 손목시계? It's not the watch that doesn't use its hands, it's him.
"heSubject watchObject hands-without removes"
The ~는 and ~를 particles take care of what is acting on what, so the No Hands bit can go either side of the watch.
I suppose that could still be "he removes the watch without its hands," but a) I'm not sure 손 means the 'hands' of a watch like in English, and b) that's an even weirder thing to say, so...
'손 없이' would be a direct translation of 'with no hands', so '손이 필요 없이' would be more accurate, since it means 'without the need of hands'