Poison damages really fast. Being able to get to a person on time to heal them seems more like chance and dumb luck than any planning or strategy.
I think it could be a cool idea if the poison effect duration was increased to give medics time to get to injured players.
Here’s an analogy: You keep a pack of tissues with you in case someone needs one. An opportune time hits and someone’s about to sneeze. Do you think you’ll be able to give them a tissue in time?