I would go with the 22-250 in either the remington 700 VSSF II or a browning m-1000 eclipse. Preferably the eclipse, since it is the most accuate rifle I've ever shot in any caliber. I would top it with a leupold vx-III 4.5x14 40mm scope or a 4.5x14 44mm zeiss conquest.
The 17 is not as good for bigger small game such as coyotes, IMO.
I like the jump in velocity the 22-250 has over the 223 so I can still push the bigger 60gr bullets down the range at 3600+ fps. The 223 factory ammo will barely push a 40gr bullet at 3600. Velocity means alot in a .22 caliber.
I figure that whatever a 223 can do, a 22-250 can do just a little better. You can buy the milsurp ammo cheap for a 223 but its only good for shooting really. I would'nt even classify a 17 in the same category. The 22-250 ammo is cheap enough. Youre looking at a 400yd rifle easy with a little practice. The 223 and 22-250 should cost the same in whichever rifle you buy.