No, to kill their bull they call upon the picdors to stab the bull from padded-horseback until the bull bleeds profusely.
Then the brave matador comes back to deal the striking blow to the now half-dead bull.
While the bull does bleed profusely, as you say, from the action of the Picadors, it is my understanding that the “Bandeleros” they place are used to disable certain muscles, preventing the bull from raising it’s head while charging. This allows the matador to better place the sword for the kill.
Disgusting sport!