Well we have certainly come a long way from the days when the choices were 270. 30-06, or 30-30. Now we have a plethora to choose from in factory loadings, not to mention what we do with wildcats. There are a lot of great ones out there but getting different opinions to agree is like lauding the attributes between Chevy, Ford, + Dodge. To each his own (+they are ALL right!)

It"s amazing when you think about the number of new cartridges that have appeared over the last few years, and I love the incredible accuracy they seem to have. Each has a minor fraction of improvement over its predecessor. HOWEVER, you MUST consider the availability of the ammo and the components for reloading as many of them "just ain't out there". For me personally the 6mm ARC sounds like the holy grail of calibers, but I have yet to see one box of ammo or one single piece of brass for sale anywhere. So as much as I admire what I read about it, I'm definitely not even remotely considering getting into it. I used to be in love with the 6mm-PPC, but who wants to pay $3 and more for brass when and if you see any for sale. There are still some old established and accepted calibers for which brass will always be available.
My advice to serious shooters is stick with an established proven caliber, start reloading, and work on finding a load that really works for you.

When varmint hunting, I generally take my cart carrying 10 to 12 rifles with me covering every caliber from .17 to .270. That way I've got all the bases covered. I do have one rifle with a cartridge that has a MV so fast that the bullet hits the varmint even before I pull the trigger~! I've interviewed many dead varmints and not one could identify what bullet from what caliber cartridge caused the end of his days.

Hope that amuses you all, and gives you as much pleasure I had when thinking about writing it

I just use a .22 rimfire or a .22 Hornet at the most!

If I want more velocity-----I JUST PULL THE TRIGGER HARDER!

Hip

Thats just like we Americans in a foreign country; when they don't understand you, we just say it louder. I don't do that but as I am an American I have to include myself. Just like the old one;
What do you call someone who speaks 3 languages?
Trilingual
What do you call someone who speaks 2 languages?
Bilingual
What do you call someone who speaks one language?
American.

No Americans speak TWO languages!

The average person speaks one!

Politicians speak another! (And I hope they choke on their lying words!

Hip

Of those choices I would pick the 222Remington
however although I own one and use it I would pick a 220 Swift after all the 22-250 is an almost as even the Ackley version just don't meet the 220 swift

I would pick a .243 as most versatile. Of those you list I would take the 6.5 . There is no wrong choice among those you listed.

.22-250 is my favorite and .223

.223 and here's why. Accurate with quite a few different powders along with availability and cost of brass. I'm practical and cheap~! No $2+ per round for this shooter.

If the availability of powder and primers were ever to return to some levels of sanity, I'd pick the 6mm PPC for it's incredible accuracy. Availability of ammo and reloading components is a very serious consideration. In a perfect world I'd say the 6mm ARC would be a real contender (very accurate and still supersonic at 1000 yards)

I built this Sako 17PPC back in the mid-1990's, to shoot 17 caliber VLD bullets at about 3900fps.
Worked pretty well with 32gr. solids.

Kevin