I think in a 20 caliber, I'd have to choose a .22-250 over a .204, or a .223. I think you can find ammo and components a bit easier for the .22-250, and it is a much better coyote killer than a .223.
I've been calling coyotes for about the past twenty five years, and have used rifles in .22-250, 223, and .243 Winchester, as well as a 6.8 SPC AR. Of all of them, I found the .243 Winchester to be the best pure coyote killer of them all. With that said, I think the .22-250 stands right there beside it. Compared to a .223, the two of them are head and shoulders above. I think the 6.8 SPC at under 150 yards is also a great coyote killer, if you want to use an AR platform carbine, it's nearly perfect.
Trouble is the .223 is a much better choice if your going to be encountering fox or bob cats, which I seldom do. Also the .223 in an AR is just too convenient and all around fine. As much as I like my .243's, .22-250's and my 6.8, I seem to gravitate back to those 5.56 AR's. It's so nice having a handy carbine, that the snow, or the rain has zero effect on. If I drop it in the mud, so what. They're just a natural born hunting tool. And if a fox comes along it won't cut it into two pieces. Also having a lifetime supply of brass for the 5.56 is a good thing. When I lose a case in the weeds I don't find myself rooting around in the grass looking for it.
Now.. Just last week I got a reminder of both shortcomings of using a 5.56 AR platform carbine for hunting coyotes. Fist was I'd set up along the edge of a gravel pit to try calling a coyote out of the brushy edge. Off to my left was an open 200 acre cornfield. After setting my Foxpro in the stalks and getting sat down against a tree, I see a black spot moving in the cornfield about 600 yards out. I crank my 3.5 - 10 scope up and see its a coyote standing out in the stalks. My .22-250, or one of my .243's whould have been a much better rifle to have along than the AR. I didn't even take a poke at that coyote. Just watched him walk over the ridge out there. A few days later I was set up in a fencerow calling to a wood lot 200 yards across a bean stubble field. I was watching upwind toward the woods when I noticed a coyote had somehow trotted in to my left, and was standing in the open field at about 120 yards. I sent a 55 grain HP into her shoulder that just put her in a spin. I sent out a second shot that took her front legs out from under her. Now she's down in the front and lets out a long howl. I send another 55 grainer up her backside and put her on the ground, and shut her up. I let her lay there while I call another ten minutes or so. When I walk up to her she growls and barks at me when I'm about ten feet away. I had to use my finish pistol on her. I only seem to have this happen when I'm shooting a .223. If I hit them in the shoulder it just won't penetrate like a .22-250 or a .243, or 6.8 for that matter. A .22-250 using the exact same bullet just has the energy to drive that bullet right on into a coyotes shoulder, where the .223 will stop, and put the coyote in a spin. If you can only have one.. Get yourself a .22-250.