I think the use of the word bow for curve or bend was used before all of the uses you mention. It comes from the word used to describe something turning back or a person taking a bow or bowing down. Bow specifically meaning bend comes from the word bugan. Where the bow used in archery comes from the word boga.

All of these do have the same origin meaning bend or curve. Specifically a bend in a river or the action of bowing. I can't find definitively if these were once separate things or always the same word.

Note the use of "arch" in archery also meaning a curve.

An archery bow is a semicircle

A violin bow is made from a curved piece of wood, the same as the weapon.