The snake is the symbol because of the serpent on Moses’ staff.
“And The Lord said to Moses, “Make a fiery serpent, and set it on a pole; and every one who is bitten, when he sees it, shall live.” So Moses made a bronze serpent, and set it on a pole; and if a serpent bit any man, he would look at the bronze serpent and live.” (Numbers 21:4-9 RSV
Centuries later, the bronze serpent was destroyed because the Israelites had turned it into an idol, in violation of The Second Commandment:
“And he did what was right in the eyes of The Lord, according to all that David [see King David] his father had done. He removed the high places, and broke the pillars, and cut down the Asherah. And he broke in pieces the bronze serpent that Moses had made, for until those days the people of Israel had burned incense to it; it was called Nehushtan.
2 Kings 18:1-6 RSV
The serpents were an idol. Medicine is an idol especially in our wicked nation.
Actually, it goes back to a Greek god of medicine.