This is not quite the case. Swords and bows have not appeared in "every culture." The melee weapon of choice in the ancient Andes, for instance, was the mace. These are represented in the art of several cultures: Moche warriors, both mythical and real are seen on ceramics; Recuay, the eastern rivals of the the moche, depicted warriors with maces in stone sculptures. Mace heads that we have discovered are typically star-shaped stone(found everywhere) or more decorated bronzes with full figures(typically Moche) or patterns(associated with Vicus culture to the north). Swords are not seen elsewhere in pre-Colombian Americas either. The closest is the macuahuitl, a central Mexican wooden staff embedded with sharp bits of obsidian, seen here in the Codex Mendoza.

A sword is a long thin peace of metal with sharp edge. Short of a club I can't think of a more common sense weapon design