Plain old curiosity. What does it offer over 223/556?
Let's take it a step farther and consider the .22 Hornet and .22 K-Hornet as well.
I grew up in western South Dakota and spent much of my adult life there as well. The edge of the closest Prairie dog town was about 150 yards south of the house I grew up in and consequently I cut my teeth prairie dogging with a .22 LR followed by smaller caliber centerfire .22s from a young age.
I was hunting prairie dogs long before it became a destination hunt activity and long before it became a commercial enterprise with dog towns being over hunted. I miss those days...
In any case, there are a few factors to consider in a varmint round:
Barrel temperature and throat erosion.
Even today however where shots are a little more scarce, a good afternoon on a dog town can involve 500 rounds, and to keep barrel temps down on a 100 degree day in the sun, you'll want to stay in the shade and swap rifles on a regular basis.
That's one advantage of the smaller .22 caliber cartridges. The .220 Swift and .22-250 generate impressive velocities, but they also generates a lot of heat and in turn demonstrate much faster rates of barrel erosion (the .220 Swift more so than the .22-250). I just don't know that many serious varmint hunters who use a .220 Swift or even the .22-250. For long shots they are more likely to move up to a 6mm, a 6.5 or even a .308 caliber cartridge, or they'll use a long, heavy .223 bullet with better wind bucking ability.
The .223 is a significant improvement over the larger .22s in terms of barrel life and barrel temperatures as the smaller powder charge generates less heat and less erosion. The .222 improves on this even further, but rounds like the .22 K-Hornet and .22 Hornet are even more barrel and temperature friendly.
Accuracy.
As noted above the .222 is a very inherently accurate cartridge, but so are the .22 K-Hornet and .22 Hornet. All three cartridges are capable of consistent sub MOA accuracy with comparatively inexpensive varmint bullets. The .223 is no slouch either, but it is slightly easier to get sub MO accuracy with one of the smaller rounds.
Range.
There is not as much difference as you'd think between the .223 and .222. For example:
.223 - 50 grain bullet
A .223 loaded with a decent a bullet having a decent (for a lightweight .223 caliber bullet) BC of .242, launching it 3,300 fps from a longish varmint barrel will have a point blank range of 242 yards, assuming a 225 yard zero and a 2" vital zone radius. At 300 yards it will have dropped 4.8" and at 350 yards it will strike 10.2" low after drifting 6.1" downwind in a 10 mph full value crosswind.
.222 - 50 gr bullet
The .222 will launch the same bullet at 3140 fps (160 fps less), producing a 256 yard point blank range with a 220 yard zero, and will strike 5.8" low at 300 yards and 11.9" low at 350 yards, drifting 6.6" downwind in the same 10 mph, 90 degree crosswind.
Realistically then it's giving up only 12 yards in point blank range, and only an inch or two greater drop at 300 and 350 yards. That difference in trajectory is less than you'll see with a 5 yard range estimation error at that distance, and if you're using a laser range finder and dope on the scope, it's not a difference the prairie dog will ever notice.
.222 - 40 grain bullet
The thing is however, that the 50 grain bullet is heavier than optimum for the .222, and it does better with a 40 grain bullet. It'll launch one at 3600 fps delivering a 280 yard point blank range, striking only 3.5" low at 300 yards and 8.3" low at 350 yards, with a still reasonable wind drift of only 6.9".
.22 K-Hornet.
The K-Hornet adds 250-300 fps over the .22 Hornet with 45 grain bullets and that weight is arguably its sweet spot. With a 45 grain spitzer bullet at 2900 fps, and a 200 yard zero it will deliver a 235 yard point blank range, striking 3.5" low at 250 yards and 9.5" low at 300 yards with 5.8" of wind drift in a 10mph crosswind. It gives up about 50 yards in effective range compared to the .223 and .222 rounds.
One of the major advantages of the .22 K-Hornet over the .22 Hornet is greater brass life as the case is worked a lot less in the firing and reloading process.
.22 Hornet - 30 or 35 grain bullet
The is where you start to see some significant differences in range potential. Newer powders look Lil' Gun work wonders in the .22 Hornet with lighter bullets and a .22 Hornet will launch a 35 gr V-max at 3100 fps or a 30 grain bullet at 3150 fps. However with a 190 yard zero, those bullets will produce a point blank range of only 220 yards, before dropping about 5-6" at 250 yards and 15-16" at 300 yards.
Realistically speaking then, 250 yards is about the maximum useful varminting range for a .22 Hornet, where it also demonstrates about 6.8" of drift in a 210 mph cross wind. You're losing 100 yards to the .222 and .223.
However, the accuracy at that 250 yard range is spectacular and you can expect to consistently place 5 shots inside a 2" circle at 250 yards provided you do your part on the trigger and in doping the wind. The long neck of the .22 Hornet results in good bullet alignment and combined with neck sizing, you'll find it's very easy to load very accurate rounds for the .22 Hornet without having to resort to uber expensive bullets.
The best part is that you can put a lot of rounds through the barrel before it heats up, and you'll be generating less muzzle report than with the much higher operating pressures of the .222 and .223.
In that regard, I like the .22 Hornet here on the east coast as it is quieter and disturbs the neighbors a lot less, and it stays cool enough that I can use it and a .22 LR and not have to worry about anything else in the more forested areas where 250 yards is far enough.
Barrel twist.
Barrel twist will largely govern your bullet selection. 1-14 and 1-16" are pretty standard in the 22 Hornet and that rules out bullets longer than a 40 grain V-max, which can be a bit too long for some 1-16" barrels - for example it will not stabilize in my 1-16" 1885 Browning, but it's a tack driver in my Zastava mini mauser that is also nominally a 1-16" twist.
I'm still a fan of 1-12" twist in a .223 Varmint rifle, although 1-9" works well with heavier bullets in the 55-62 grain range. The problem is that over stabilizing the bullet with excessive rpm causes problems with less expensive, less precise bullets where the center of gravity and center of form are not on the same axis.