Should we allow certain species to become extinct for the sake of environmental integrity?

Do we arbitrarily allow the demise of some species based on some criteria that a handful of people enact? Who decides what stays and what goes? The requirements for sustainablity vary radically. Environmental integrity? Kill kudzu, the cane toad, rabbits and goats on various islands that have fragile ecosystems, but don't let the poorly studied weak organisms p from the world without knowing if they had the cure for cancer, or something like that.