These three verses together encapsulate the entire great controversy in the cosmic conflict between the serpent and the Lord. Let's look at the original story, to which Jesus was alluding. Remember that sin entered the world when the serpent succeeded in tempting our first parents to doubt God's word. And Moses prayed for the people. For this nation of shepherds, the serpent lifted on a pole served as a strong symbol which they all well understood.
Snakes are a deadly menace to sheep.
A dog might get bit by a rattlesnake and survive without any special treatment, but sheep are much more fragile. This is one of the reasons why shepherds must carry a rod. When I lived in the desert mountains, my snake stick served a couple of purposes. But a mortally wounded snake might continue to thrash and writhe for hours. So rather than grab it with my hand, I would lift it up with the rod to move it far away from my premises.