If you're talking 500 yards, the .220 is the thing for coyotes. It's not really noticeably more wind sensitive than a varmint weight .243 bullet. Barrels should last 3000 rounds, a lot of coyotes.

At longer ranges a .243 or larger caliber will typically be less wind sensitive. In benchrest very few shoot a .223 caliber beyond 200 yards. An extremely accurate cartridge that is capable of shooting well beyond 300 yards and still be accurate enough to be effective on crows is a 6mm Benchrest Remington / Norma. The next up typically is the 6.5-284 Winchester or a 6.5-06 standard or improved variant. Any GOOD Benchrest gunsmith can be a tremendous asset here.

I have a 220 with a Shilen Bbl on Mauser action ,with Timney trigger, 4-26x40 target scope, I have shot one chuck at a measured 570 yards, using a bipod. I have also used it in running deer competion, I have 1000= rounds thru it , so far so good. I ahd problems early on with case's and now use only brass case's, no nichel they needed trimming after one loading. It is a great round enjoy.

Get a Swift,my favorite caliber.

I have a 1976 Ruger Varm. that has 1000"s of rounds thrugh it and it still shoots 1/2 in. at 200yrds. If you don't get it to hot it will last a life time. If you load it real hotit only does what any rifle will do. Now I've been at it for over forthy years, and even 22-250 are hard on barrels if you load them for all there is.

Mikehjd: I presently have 3 220 Swift Varmint Rifles on line. It is one of my favorite calibers. Lots of great bullets out there and powders to make them perform very well indeed.
Keep an eye on the barrel temperature during Gopher shooting sessions and it will "live" a lot longer. I had a close friend who was so impressed with one of my Swifts he bought one just like it - a Remington 40XB in 220 Swift. He made the mistake of taking his Swift along on a midsummer Prairie Dog Hunt in Montana. The leades of the rifling literally were gone for 2" up his barrel in 1,800 shots. His mistake was firing the Rifle repeatedly while hot! He sent it back to Remington and they rebarreled it for him ($400.00 charge!). I tried to talk him out of it at the time as the Rifle was still shooting close to 1/2" groups at 200 yards. The new barrel did not shoot that well. He just had an extremely long throat on the first barrel. Lesson here which I have seen from time to time is having a long throat does not automatically mean poor accuracy. Often but not always. Your 17 Remington and 222 Remington would sure do well in the Gopher fields so maybe save the Swift and the extra powder for Coyotes.
I use my swifts for Prairie Dogs, Rock Chucks and Coyotes mostly. The significant extra range, flatter trajectory and less wind drift with the Swift over the 222 family of cartridges is worth it to me. But once the barrel is warm I switch guns and let the Swift cool. Shooting the Swift with a hot barrel is certain to wear the barrel a lot faster than shooting it cool.
Good luck if you decide to try one!
Hold into the wind
VarmintGuy