I think it's pretty simple - if you don't see a reason and don't have a desire to hunt coyotes, then don't hunt coyotes.

I can say with confidence that you can hunt enough to impact your other game populations, but that's really just by applying pressure and driving them out, not really by eliminating population. Equally, I can say with confidence that you'll never be able to decimate their population to create an imbalance in their "cycle." If there's food, coyotes will be there.

If you're a die hard turkey hunter and want to foster your turkey population, then hunting some coyotes will help. You'll also likely be surprised what it does for your deer herd, even though you already have a reasonable population.

As for your dog - a large dog - there's really only one time of year where such a big dog would be at risk, unless it goes running off into the night chasing a coyote. In the pre-breeding and breeding seasons, males get very territorial, so they're more apt to come near homes and attack large dogs. Small dogs are a snack any time of year, but big dogs usually aren't a target except for the late Dec through early Feb timeline, and dawn/dusk/night are the high risk times even in that period.

So again - there's a few "if's" for you to consider - if you're a die hard gobbler hunter, or don't want to manage your dog's outside time during the late winter and spring, then you'd be motivated to apply hunting pressure to coyotes. If you're not, then why would you hunt something you don't care to hunt?