I think it matters whether you just want a dead coyote, or you are hunting for fur.
I was hunting coyote for fur, before there was a fast twist .223 and heavy bullets. When the lightest .243 bullet you could get was 85gr.
And, before there was an Internet.
When the factory .22-250 load came out with a 40gr bullet (that's right a light bullet, NOT a heavy one) local dealers couldn't keep them on the shelves, the pelt hunters bought them all. Why?
Because that light bullet, at that high speed was not only a "lazer death ray" but also because, generally, the bullet completely fragmented inside the chest cavity and (again generally) did not make an exit hole.
ONE hole in a pelt is always better than two!
the exit wounds were about the size of a quarter.
That being said I did shoot one with the above load that was about 200 yards from me, the exit wound on that one was about the size of a baseball.
This is fine performance for killing the pest, but miserable from the viewpoint of the pelt hunter. Nothing like turning a $125 pelt into a $50 pelt to convince people to use something else.
Of course, that doesn't matter so much today as the fur market tanked ages ago.
As much as the cartridge makes a difference, so do the rifles. Sporter weight guns are seldom in the same accuracy class a dedicated purpose built varmint rigs. How much do you need, vs your hunting style is always a trade off. A 7-8lb gun is a different proposition than a 12lb+ gun, at least for me.
The larger the round, the larger the powder charge, the heavier the bullet, does matter, but does it matter enough, for what YOU are doing?
I have .22Hornet, .221 Fireball, .222Rem, .223 Rem, .22-250, .243, 6mm Rem, and .25-06 for cartridges I put in the "varmint +" class.
But not all my guns are good varmint guns. My .243 and 6mm are carbines, (stalking rifles), and not as good for long range varmint shooting as a different rifle in the same caliber would be.
All arguments for "less powder, cheaper brass" etc., have to be balanced against the performance you are looking for. If you want performance, you HAVE to PAY for it.
More expense in materials (including barrel life) is the cost of higher performance. If a .223 does everything you need, use a .223. If you need (or just want) more, you are going to pay more for it. Accept that, and move on.
If you want the ultimate (practical) long range varmint round, skip the 6mms and get a .25-06. The effect of an 87gr HP pushed by a case full of IMR 4350 on small game has to be seen to be believed!
Yes, its going to cost you more money every time you pull the trigger. But nothing else delivers quite the same performance. There is no free lunch.
Since I have a fine M70 Varmint .22-250, and a 1903 Springfield someone turned into a beautiful varmint rifle in .25-06, I've got no need for a varmint gun 243. Both these rifles will shoot 3/4" IN MY HANDS, with the right loads, and that works just fine for me.
Every rifle is a bit different, and while round A might beat B on paper, it might NOT beat it in usable performance from YOUR rifle. For example, a 55gr .243 beats the 55gr .22-250 on paper. Now what does that round do in your gun in the field? Maybe fine, maybe super, maybe not so much. If you rifle only groups that 55gr .243 into 2" (or worse??) and your .22-250 throws into 3/4 or 1/2", which is actually superior?