It has occurred to me that another scenario could be the development of acupuncture from tattooing. Tattooing has been practised in most pre-technological societies and may indeed be a near-universal human activity, going back to Neolithic times or even earlier. It doesn't seem unlikely that it would on occasion have provided coincidental relief from pre-existing pain. Moreover, magic diagrams may have been tattooed specifically to cure disease and this would have sometimes worked.
Over time, the diagrams would have become more perfunctory, gradually degenerating into simple lines; ultimately the tattooing would be dispensed with and the needle would be used on its own.
Possible supporting evidence for this idea comes from the discovery of tattoo marks in the lumbar region of the so-called Ice Man discovered recently in the Austro-Italian Alps, who died some 5300 years ago. It has been suggested that these marks were made as a treatment for back pain. If so, this would be an instance of therapeutic tattooing performed in Europe in a remote era; a similar practice could have arisen independently in China. The so-called acupuncture needles found in tombs in China could equally well be tattooing needles.
In this connection I'm grateful to D. Moyshe Kalman for the following comment.
If, in antiquity, the tattooing was performed as a ritualistic act initially to heal a problem, such as a shaman or priest tattooing a bad knee with a symbolic picture to attract some healing power, the initial act would have been perceived as therapeutic and over the years could easily have led to the invention of acupuncture. Tattooing is absolutely forbidden by the Hebrew Scriptures as a heathen practice, which indicates that in antiquity it was viewed as a ritual act rather than a cosmetic act. Body piercing in the nose or ears is not forbidden and is seen as a purely cosmetic act. I think this strongly confirms your link with tattooing and the origins of acupuncture.