the animal is killed and then instantly has its throat cut to bleed it.
In non religious slaughterhouses, hopefully the animal is stunned so that it doesn’t feel pain and then bled. In the US, for example, cattle is shot in the head through the brain with an air piston device. In Bible times, however, there were no modern stunning methods. All they had was a knife and cutting the cartoid artery with a sharp knife rendered the animal unconscious before it was aware of it as it bled to death. That was the most humane method at that time.
Today, for production line slaughtering in religious slaughterhouses that is very time consuming so they use another method to secure the neck so they can cut it, which causes pain to the animal as the animal is struggling.—Romans 8:22 But the Jewish slaughterhouses, I believe, have a higher standard for healthy animals and I believe they sell below standard animals to gov inspected meat plants.
Another way to look at this is that being killed is painful any way. Making it less painful, the animal still suffers.