Really you need at least two rifles. Coyotes and white tail deer and some other critters a 243 or 7MM08 is ideal although the 7MM08 is border line for coyotes if you want to sell the hides and don't like to sew. Better if you hand load.
Elk and shooting the long distance the Winchester 300 Mag does fit the bill. Nag Na Port it to reduce the recoil to 243 levels with out the back blast of a brake.
You can reload a 110gr. varmiter bullet for it that may not tear up a coyote hide to bad.
Only way I see one rifle to do the whole list is to handload.
Al