I think as big of a factor is the aggressive doe harvesting. Think about it - if coyotes or wolves on their own caused the collapse of deer populations, by the time humans got on the scene the deer would be extinct.
So here we have two issues - aggressive doe harvesting and coyotes. The one we have absolute control over is doe harvesting - do what we do in ontario, and have people apply for doe permits. We've had a problem here with a lot of does getting harvested when the deer population was booming, but people thought the deer population could take it. one bad winter later, combined with overharvesting and high coyote populations, and the deer population, while still OK in my opinion, is nowhere near what it was 5 years ago.
The lesson here is this: you can shoot as many coyotes as you want, but there are always other factors than predation. Predation is often the tipping point - the straw breaking the camel's back. But it's much more difficult to deal with than permit issuing policy. Anyone ever try to keep coyotes down through culls or hunting pressure? How'd that work out? Some studies I've seen indicate that coyotes respond to mortality by having bigger litters.
The cure for coyotes? A healthy wolf population. Go anywhere where wolves used to rule and where they've been extirpated, and coyotes have moved in. Then, of course, you have wolves competing with humans for deer.
If I was game management there (I'm being an armchair biologist, I know, but I did go to school for this stuff) I'd be encouraging coyote trapping and hunting, temporarily reducing doe permits and doing some serious necropsies on any dead deer I find to determine cause of death. It's a little more complicated and less popular than trying to kill all the coyotes, but since its virtually impossible to eradicate coyotes, why try? Just take a little off the top and hope for the best.
Besides, why do we really dislike coyotes? I guess they're just too good at their jobs of hunting and killing stuff.