hunted 1/2 my life with bolt guns , the next 1/2 with both AR platforms and bolts .
For coyotes and hogs , AR all the way if your baiting hogs or calling yotes , many times there will be multiple targets . 6mm arc would be my choice from the cals u listed.
i personally use the 6.5 grendel as i have many 6.5 cals and a shelf full of that cal bullets to hand load for.
i run a 95 gr Vmax and coyotes just plain dont get back up from a hit .
223 they can spin and run and shake off the hit needing fast follow ups. Shot placement is key to anchor them with 223 head ,neck , hi shoulder works perfect.
if your saving pelts then smaller, light , varmit bullets are key .
for me im saving my livestock and could care less about a pelt thats worth $6-$8
If your just shooting when you see them hunting around your property , then a bolt gun is a little cheaper .
A custom built accurate AR will set you back a few bucks but very worth it if your shooting ranges are extended. Coyote have alot of hair which makes then look bigger than they are .
im not gonna suggest hunting at night as the budget to do so will scare the most understanding wife into a rabid encounter !!
you have been warned ,
PS
thermal , NV , caller, and other gear for night hunting cost is expensive
and SO WORTH IT !