Remington ADL synthetic stock in .30-06 with Leopold 40 mm 3.5-10x scope in Leopold rings. Total price $330 rifle+ $500 scope+ $50 rings + $20 rifle sling is about $900.

The .30-06 is good for anything in the lower 48 states; if you are going to go to Alaska, buy an additional gun. If you don't accept this finessing of the problem you posed, then change to the .338 Winchester Magnum which IS suitable for all game including Alaska Brown Bear, but I think it is a bit much for pronghorn antelope and whitetail deer. You can buy ammo for the .30-06 anywhere.

The Remington ADL probably shoots as well as any other rifle you are going to buy in the price range you mentioned. The synthetic stock makes this rifle substantially impervious to the elements.

The scope is very good and will stand up to abuse and provide good service. Is it odd to invert the budget for the rifle with respect to the targeting system? Not according to my viewpoint. Any damn rifle is going to put 3 shots inside a 1.5" circle at 100 yards if you do your part. If your scope breaks or fogs or is unreliable, however, even a $4,000 rifle is not going to shoot where you want it to.

I actually own this combination. I bought it as a back-up for my other rifles. One I can take out when it is raining and I don't want to expose my pretty walnut wood stocked rifles to the rain. One which can provide a backup to either my .243 or my .25-06 on an out-of-state pronghorn hunt; one which can provide a backup to either my pretty walnut wood stocked .30-06 or the .338 Winchester magnum (I may someday purchase) on an out-of-state elk hunt.