Man, I didn't read through everyone's replies, but as far as I can tell you want the following:
TIKKA T3X CTR (Compact Tactical Rifle) in 6.5 CM. It is within your budget at around 1000.00. It is INSANELY accurate with factory ammo. Absolutely incredible. It is very LIGHT WEIGHT with the factory composite stock (vs. the heavier chassis on RPR, etc.). It comes with a factory-installed rail (figure another 200.00 into your budget for a rail if considering a rifle without one). For your purposes, figure on adding a brake, and possibly a stock pack to bring your cheek weld up a bit, and possibly a Harris bipod if you need one, and you're ready to go. I have the 20 in. version,. It's a 1:8 twist, and it's good on IPSC targets with factory 140s at least out to 1200. I see where they now have a 24 in. version out. If I only had one rifle, I might go that route, but I already have long-barreled custom rifles, so I like having this little compact, light-weight 20 in. hot rod for something a little less trouble to lug around. It runs factory Hornady 140 ELD-Ms at 2690. My 26 in. Krieger-barreled custom rifle runs them in the high 2750s for comparison.
Why not shoot a coyote at 700? The rifle is more than capable, at least. I routinely make head-shots on an IPSC target with mine at 700 (bipod and rear bag-supported). I personally wouldn't trust the horsepower for an elk at that range, though some might.
Get one, it is exactly what you are describing you want! Mine shoots factory ammo just as well as my 4500.00 custom builds. I think it will most likely out-shoot everything else you your list.
Edit to add: I tend to agree with the last two posts above IF you truly aren't going to ever be shooting beyond 4-500 yards. I might go .308 in that case, and have a little heavier punch on deer at shorter ranges. But if you can only have one rifle, and you want the capability to stretch out when the opportunity comes, you are on the right track with the 6.5. You can get the T3X in either cartridge.