Because people are afraid of guns. Because coyotes can be tricky to shoot when they are moving and most people are not good enough shots. Because coyotes may not be out during the times that a person is around to find them attacking their animals. Because people are stupid.
Many people use poison for rodents. Other animals eat the dying or dead rodents and are then poisoned themselves - including people's pets who eat the poisoned mice and the owners can't figure out that they killed their own pet.
They just don't stop to think about the long term ramifications of their actions.
I would rather shoot the 'yotes. Though, I did see one rancher use snares once, to solve his 'yotes. A lot of dead 'yotes all the way around the perimeter of his fence line in various stages of decay.
That's kinda nasty. I'd at least bury or burn them rather than let them sit around and rot. We shoot them here too if need be.