It’s not for sport, it’s for pest/invasive species control. The fact that you CAN eat them, should be a bonus for people looking for free meat. I have never had nutria so I have no idea how much of an uphill battle that is.
The dead rat toss is a little macabre, but you have to understand how bad invasive species are on local ecosystems. They are a scourge that is destroying local habitat and local populations.
Absolutely feral cats need to be culled in some locations. New Zealand, Australia, and Hawaii have had feral cat culls to protect their endangered bird species.
Look, I love pets and I like animals in general, but if you don’t have a balance then species go extinct or have their ranges reduced. There are so many example of plants and animals where “God didn’t put them” (as my dad would say) that are killing through predation or out competing native species.
Yeah, it is an uphill battle. I doubt you can ever eradicate them, but you might be able to contain them.
I read an article on pythons in Florida, and holy hell. These snakes are so good a hiding even the ones they have trackers on they can’t find sometimes. And they just devour anything they can fit in their mouths. 
Ironically they would help keep the Nutria population down, but introducing them to Louisiana isn’t the answer. (I am sure they have some down there too!)
Another invasive species stories is Grey Squirrels pushing out native Red Squirrels in the UK. The grey ones have a higher tolerance for tannins than the red ones, so they can eat acorns sooner than the reds. Imported in the 19th century from North America for gardens, they quickly spread with the help of humans, hitching rides on cars and such.
