The recreational hunting of coyotes typically only serves as a short term reduction in the local population. One study in Georgia suggests that you may benefit 1 year of fawns however recruitment was not affected. Transient animals will very likely fill in the void in available habitat or females will go into estrus faster/more frequently. Not discounting a good coyote hunt since those can be fun and a great way to spend time on your property
"We reached the old wolf in time to watch a fierce green fire dying in her eyes. I realized then and have known ever since that there was something new to me in those eyes, something known only to her and to the mountain. I was young then and full of trigger-itch; I thought that because fewer wolves meant more deer, that no wolves would mean hunters’ paradise. But after seeing the green fire die, I sensed that neither the wolf nor the mountain agreed with such a view.”
Also Leopold. Thinking Like a Mountain, A Sand County Almanac.
Those who explain why they like to shoot coyotes is that they like to shoot coyotes - they're stating a fact.
Most other reasons stated for for liking to shoot coyotes - interesting talk over beers at the Longhorn/Moe's/The Mint/Karson's Streamside Brew Pub and Tapas (for the flat bills), conversation gets "better" with more beers/ales..........