When the actual culprit is caught and killed, people see that the job is already done. Singling out the killer for execution drives home the fact that man-eaters (or man-killers, in the case of Steve Irwin) are the exception. When a killer is left out in the wild, the accompanying message to the public is something along the lines of: This is just the animal acting according to its nature.
It is undeniably the nature of predators to hunt and kill. But the scarcity of attacks on people suggests that hunting humans is not normal behavior among predators. There have been 225 documented cases of major bites from alligators in Florida since 1984. That number is remarkably low for a state that currently has a population of roughly 1 million alligators and 19 million people.
Man-eaters do have a tendency to turn a one-time thing into a habit. A small number of aberrant animals are responsible for a shockingly high share of human attacks. The infamous Panar leopard of Northern India killed and mostly ate 410 human beings until the famous hunter Jim Corbett ended its career in 1910.
A single, huge crocodile in Burundi had eaten up to 300 people as of 2008. ”Gustave” is presumably still out there in the wild. A Frenchman by the name of Patrice Faye spent about 15 years unsuccessfully trying to trap the creature alive.
The case of Gustave illustrates the problem with a “die and let live” approach to man-eaters. In spite of the massive body count racked up by this animal, both Faye and a group of biologists, including herpetologist Brady Barr, spent quite a long time trying to capture Gustave alive rather than simply kill the croc. Their goal was to advocate for conservation of Nile crocodiles, but as Barr himself pointed out in National Geographic:
"People have to get their water, do their laundry, fish for a living," says Barr. "If a croc does take a person, villagers may slaughter a few crocs after an attack—enough to feel as if they've done something—and then they go back to doing what they have to do."
How many innocent crocodiles were killed in retaliation for Gustave's attacks while Barr and Faye were messing around with their cages and snares rather than killing the crocodile? This wasn't good for either the humans or the crocodiles.
These repeat performances are typical among man-eaters of many species. Bears, lions, tigers, leopards, alligators, crocodiles, cougars. Possibly sharks as well, assuming the 1916 attacks that inspired Jaws were in fact the work of a single shark.
The man-eater is exceptional. It isn't a normal predator. The idea that the man-eater is an innocent totem of nature while man is the guilty interloper simply does not hold up to scrutiny.
Unless the species' numbers are so low that genetic diversity is in immediate danger, there is no advantage to letting an animal like Gustave live. The consequences of leaving a man-eater in the wild, whether it is the brown bear that devoured Timothy Treadwell or the gator that swam off with Kaleb Langdale's arm, are terrible for nearly everyone concerned.