farmers, whose livestock and poultry are terrorized by coyotes and foxes, sometimes mountain lions.
I don't think farmers regularly take potshots at coyotes or something, especially big corporate farms, and that's really cruel to the coyotes and foxes, who are just trying to live.
For hunters, who keep animal populations in check
or extinguish them - and what does it mean, "in check"?
use the animals to feed families
uh no, that doesn't happen, in America, anyway. Maybe just for novelty of it, but it's 100% for fun. Furthermore, that's also extremely disrespectful to eat those who you have killed - as if it wasn't cruel enough to kill them, even if it was somehow needed.