I tend to agree that you're not going to get much rifle anymore for $500 if you're buying new. I don't know that a bull barrel is an absolute necessity, though.
I've found myself not to be a raving fan of the .22-250. It's usually marketed as a varmint cartridge and packaged in a "varmint rifle" - not something really well-suited to clunking around after coyotes on skis or snowshoes. Further, it's draw often seems to be centered around that blistering 4000 fps, 40 grain bullet. I've found that behond 300 yards, that little pill sheds performance at an alarming rate. So much so that I'll take a .223 with a 60-gr bullet over a .22-250 with a 40-gr bullet any day I expect to be shooting beyond 300 yards. I'm happy with the .22-250 now, but only now that I've shed 200 fps by moving to 50-gr V-Maxes. It's weight is still a problem, thanks to a 24" bull barrel and varmint stock, and it's best suited set up over a prairie dog town. I might be tempted to use it on coyotes if it weren't so heavy.
If you can find a .22-250 with a sporter or simliar taper barrel, it'd probably be fine as a walking coyote rifle. It'd be better loaded with heavier (higher BC) bullets. And then there's glass. Varmint rifles tend to get those 50mm objectives and 20-plus power on the top end Hubble telescopes. Not useful on a walking coyote rifle.