Shooting coyotes doesn't get rid of them. If anything, it creates more coyotes. If shooting them was an effective way to control their population then coyote numbers would have plummeted like wolf numbers did in decades past when predators were shot on sight without prejudice - instead the coyote expanded as the wolf was shot out.
If you want the pelts, blast away! If your only motivation is to help the deer herd, there are many far more effective things to do. (btw bears are devastating on fawns, but you've got limited tags on those).
That's what happens when hunters just shoot at the pack or individual animals indiscriminately.
When both of the Alpha/Beta pack leaders are killed, then Coyotes tend to over populate.
I've seen Coyote #####es put the run on young males, trying to cover young females, especially when the populations are high and food is scarce.
I would say population control is the main reason they do this but it also may to to stop inbreeding? which would likely be more instinctive than anything else.
The reason I came to that conclusion, I've seen the Beta females stop the Alpha males from covering different females in the pack, for no apparent reason. Inbreeding is about all I could see at the time.
On a couple of occaisions where there are a couple of packs in an area, say around a large Dairy Farm or even a Village, such as the one I live close to, there will be several different packs that very seldom interact or even cross over each other's marked boundaries.
Around this time of year, the males start to wander out of marked boundaries, looking for receptive females, which the Beta ##### will sometimes tolerate.
Shooting every Coyote on sight, without identifying its status in the pack will definitely lead to overpopulation.
I can understand why a fur hunter/trapper wants the larger hides, if they're in prime condition, like they should be right now, but even the experienced ones know better than to overdo a good thing.