I never shot a coyote however hundreds of fox. Even on the smaller fox I preferred a slightly stronger bullet construction, like something used for Roe deer. My 223 with 50gr hollow points I often had fox not being killed properly if shot from behind. Fox control means shooting centre mass from any angle for me. 69gr in 223 worked very well but the bullet is quite slow out of 20". Not great for quick shots under the lamp at distances up to 300m. The 22-250 with 52-55gr is great for under the lamp but not as good as 69-75gr from a 223 at longer range. So at the moment I have a 22-250 with 1/8 and use 75gr ELDM (3150fps) which is just about perfect in performance, shot fox out to 600yd and today a crow at 480yds in 13mph wind. Fast twist 22-250 are mostly custom jobs and OTT for most, leaving the 243 or similar as an easier choice. For around 2-3 years I used a 20" Remmy 243 with Sierra 80gr SPBT (~3200fps) which is a fantastic bullet, expands well but is not explosive and can also take down Sika Stags in the rut.
One thing though with all of these cartridges, measure muzzle velocity when selecting factory ammo. Some ammo is shockingly slow. I have measured around 2800fps for Fiocchi 50gr Vmax 223, Sako 55gr around 3100fps from a 22-250 22", Sako 70gr under 3000fps from a 243. A well loaded 223 mit be similar to some factory 22-250 loads and not that far off a very slow 243 factory load.
edi