There is no way to have a natural balance right now. Naturally large cats and wolves would control the population of coyotes and thus fox etc ...
Without wolves and large cats coyote populations are likely to boom and bust with prey animals populations.
So as coyotes move into new territories and their population rises we will see a change in deer and other prey species to achieve a new balance.
If man goes in and kills off all the coyotes then the burden is once again on man to control the deer population either by hunting or deer vs car collisions or watch populations boom and bust with annual rain and weather conditions that affect food supplies.
I am not really interested in coyote hunting and our deer herds are larger than ever here in my area. Coyotes are definitely on the rise I believe.
The predation will likely strengthen deer herds but hunting practices should evolve in light of coyote predation.
this is just my uneducated logic here... I am no wildlife biologist.
Now lets talk about squirrels...
(how to get all the owls and hawks to come in and take out a few more of these pesky tree rats)
I thought about building little platforms around in the forest with tiny bottles of ketchup, and BBQ sauce for the hawks and owls with some arrows that point and say "Squirrels are here"