I think the snake on the cross comes from the greeks, not the Judaic influenced christians of Israel and Judea.
If you did a little research you might be surprised... The serpent and the tree itself create the caduceus symbol. The cross is the symbol for a sacred "medicine." Which is why Moses puts the serpent upon the staff in that myth.
Most of the Old Testament was influenced by Egyptian Mythology/Mysticism, who also use the caduceus throughout their symbolism. I've also seen depictions of the serpent upon the Ankh/cross in their ancient art.
The New Testament is more based upon Greek Mysticism/Mythology who based their system upon the Egyptian System also but also is heavily influenced by Zoroastrianism, Vedic Hinduism, The Kaballah, and the Greek gods Apollo the solar deity, Dionysus/Bacchus, the Shaman, and Prometheus the rebel. Which combined into one become the Jeesus, the magic man, the shaman with his magical plant spirit. The self realized god man who is the sun upon the earth and the sun up in the sky and the enlightening drug itself. Eat of his flesh... and go to outerspace/heaven.
In the Vedic Hindu Rig Vedas they had a drink of "eternal life," which was made from the stalks of three plants and personified as a deity.
Christians have basically borrowed and stolen a bunch of ancient myths they have no understanding of and turned them into a mountain of poop. These stories were intended for initiated members of ancient secret societies or mystery schools, not the uninitiated profane ignorant masses. These mystery schools were very popular in the "holy land" and through southern Europe and India at the time. Whenever they find really old temples and claim they're "Christian" temples they're actually the temples of these Gnostic mystery cults.... like the Essenes, whose leader Philos knew the secrets of the sacred pharmacia.

. Northern Europeans had their own versions of this astronomical solar shamanic belief system as did most other people all around the world including the Aztecs and Native Americans. Then came Churchianity and Roman conquest to enslave the planet with their backwards mythology known as Christianity.