Why would you pick the 204 or 22-250 over the 222 or 223 for that matter?
Speed, trajectory, recoil, noise.
If I were keeping the pelts then I'd have to look for a bullet that didn't do too much damage. Although I suspect that you'd find that the 32gr 204 at 4200fps would do just fine. Even on woodchucks I frequently get no exit wound. I would think that on a coyote you almost never would. The 22-250 would have similar options to the 204.
On recoil, it's not like 222 or 223 are going to have high recoil, but the 204 generates very high energy levels with ZERO recoil, for all intents and purposes. I mean, it barely twitches.
Trajectory, compared to the 40gr 204, only the 22-250 and a couple of other cartridges with HOT handloads can match the 204, or even come close in most cases.
Noise, the 204 is quiet **for a center-fire rifle**, not anywhere near *quiet* but quietER. The 22-250 is definitely louder. The 223 is quite loud also, although I don't have as much experience with that round, it is noticeably louder in the few rifles that I've been around. The 222rem, I've never been around. It seems like it might be similar in noise to the 204, but that's just a guess.