The only way to kill a person who had ingested that potion was by beheading him/her.
My understanding is that the Substance doesn't have any effect after it being used. A member of the Hand can be killed just as easily as any other person (other than, of course, the fact that they've spent centuries training in the martial arts, have extensive criminal organizations that supply them with bodyguards, and, at least in the case of Gao, have supernatural abilities). You don't need to cut off their head to kill them, you need to cut off their head to prevent them being resurrected.
During centuries, the Elixir granted the leaders of the Hand with an incredible longevity. When one of them was killed, the "fingers" of the Hand also used the elixir to bring them back and repair their bodies.
Given the wording "repair their bodies", my understanding is that limbs and organs can be regenerated. And of course any organ is going to suffer damage from being deprived of a blood supply, so the Substance needs to be able to reverse that.