We found here when hunters wiped out the coyote and fox populations, the rodent populations went right through the roof. We had severe rodent problems for years, until the predators regained a foothold. For this reason alone, we don't kill predators but build to deter. In 12 years of keeping up to 90 birds a year, we've only lost 5 to predators.
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I don't doubt it. Coyotes and foxes have a really interesting relationship. They can co-exist except a coyote will go out of its way to kill a fox. When you add the longer travel distances of coyotes, a lot of studies show that very aggressive wildlife management programs result in only seasonal declines of coyote populations. The following year you can end up with MORE coyotes due to an abundance of prey species, as you note, and increased coyote litter sizes due to a lack of coyotes in the territory. And red foxes can have a hard time getting a foothold when the landscape is saturated with coyotes--this is why you see more foxes in suburban/urban areas.