I'm retired and I live in Montana. Add those two together and you can get a lot of dead coyotes. I've hunted them for 55 years, and have used at least 15 different cartridges during that time. There is no single "best" cartridge for coyotes. There is simply too much variation in terrain, and distance, from one part of the country to another for that to be possible. There are some general things that I have learned over the years that are true that might be of benefit in answering this question.
Bullet weight is important! I built a wildcat similar to the 20 Practical about 10 years ago when fur was still worth a little money. Everyone raved about the 35gr. Berger as being a great bullet on fur. In my experience it was a piss poor coyote combo. Very inconsistent performance with one shot performing perfectly, with it being almost impossible to find the hole, to the next shot yielding a surface blow up with a hole the size of a soft ball and a running coyote. I haven't used for coyotes since then.
The .223 will work well within its range limitations. If you live in an area where the majority of the shots are less than 300 it will do the job with the right bullet. I've seen far too many failures from the .223 at distances past that.
By far the most coyotes I've shot has been with a .22-250. Well over 1k. The 50 grain bullets work well out to 400 yards or a little better and the larger bullets will extend that range a bit further. The bigger case capacity .22's are a great choice if you hunt in the part of the country where shots can get very long. And that isn't limited to "the west." I hunt in central Kansas every year and there can be some long shots there.
I put in quite a few days in northern Montana near the Canadian border calling coyotes, and my choice for the often long shots there is a .22 Creedmoor. It's a stellar performer in that terrain. I like the 69gr. Lapua Scenar bullet for it extreme accuracy at ranges out to 600 to 700 yards. The longer, higher BC bullets don't start to shine at most coyote hunting ranges.
The bottom line to all of this is shoot what you like, but figure out the limitations of the cartridge you've chosen and compare that to the distances you will likely be shooting.
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A prime Nebraska Sand Hills coyote shot with a .22-250 and one of my 52gr. hand swaged bullets at about 250 yards.