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Shooting accuracy is about physics, not magic
Actually judging what a cartridge will or will not do is little more than common sense mixed with the laws of physics.
For instance, it has been said to me that the 5.56 military round (.223) is a better man stopper than the .308. Now I have shot many a 10-pound groundhog with this round in its .223 guise and it is not reliable at anchoring hogs at ranges in excess of 200 yards. This does not make it a poor varmint round but remember a groundhog is a far smaller animal than a human.
There is one caveat in this equation and that is that the military round is non-expandable while sportsmen use soft point or hollow point ammo when hunting varmints. The arguement then says that the military bullets tumble on impact. Logic and laws of physics tell us if that were so then the bullet is marginally stabilized and accuracy would be questionable. There is no magic.
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A case that is large for a small diameter round like a .22 can be medium sized when the hole in the barrel is expanded to .30 caliber. And when caliber and pressure are the same, the larger casing will produce more velocity.
A Factor such as barrel length is related to firearm not cartridge.
You can reach a point when moving to larger cases cause diminishing returns and ask yourself is it worth it?
For instance, a 6mm case holding 50 grains of powder may achieve a velocity of 3300 fps. Moving to 60 grains of powder speeds things up to 3400 fps and cut barrel life in half.
Now take a 6mm case holding 70 grains of powder and you may gain 50 fps but cut barrel life to one quarter of what would occur with the original 50-grain case.
That is diminishing return at a high price in barrel life and powder use. I know some have owned a .30-06 that was slower than its little brother, the .308. In that case it was related to the rifle.
Guns do vary in performance within the same cartridge. Or it can be that one is loaded to a higher pressure than the other. We do not own equipment to measure pressure.
As far as rifle variation, I own a few rifles chambered to .270. With the same person loading, the same press, balance, box of bullets, primers and powder, there is 200 fps difference from the fastest rifle to the slowest. You can't keep things any more the same and yet there is that large difference on the chronograph. You explain.
So with that in mind there is no question that the .222 will not - on average - reach the speeds of the .22-250. Nor will the .22-250 move a bullet as fast as the .220 Swift.
I once chronographed a .222 that was moving a bullet faster than another shooters .22-250. But that was because the .222 was loaded to a much higher pressure.
When the same bullet is used and the caliber is the same, the larger case has the higher velocity potential.
The reason for the high velocity being desirable is the faster the bullet moves, the flatter the trajectory.
The flatter the trajectory, the less one has to guess distance in the field.
Why the existence of the smaller rounds?
For one thing they are less noisy around houses and other inhabited buildings. The more powder that is burned the louder the gun.
Another plus to the smaller round is that it is more economical to shoot. With today's component costs this becomes more important.
Another factor is accuracy - that important ability of the rifle to place one bullet in or near the same hole as the previous shot.
It has been proven that little cartridges shoot more accurately than larger ones. If you don't believe that statement check out what is used at a bench rest match.
We learn to compromise wanting adequate accuracy coupled with high speed.
There is no magic.
