If you want to get at why a snake was used in Genesis, its illuminating to look into what snakes meant in Near Eastern culture of the time. Mesopotamian culture thought of snakes as immortal due to shedding their skin. There is the famous Epic of Gilgamesh which features a trickster snake stealing the plant of immortality. The Epic of Gilgamesh is much older than the bible and was well known, and there are a lot of resonances and parallels with the Genesis story, with the tree of everlasting life and the flood story.
The Sumerians apparently had a snake god, and bronze age Canaanite culture from which the Israelite culture grew out of, had a snake cult according to the wikipedia article i just looked at. The bible fable can't be taken in isolation, it is using the imagery of the snake with the associated meanings of the time that would have made sense and taken for granted by the people in that time and culture, and resetting it in a new context to make a point.