Most off the shelf rifles can shoot 1 MOA, some less. If you want something that will shoot right out of the box make sure the thing is free floated as a start. I believe all savages are supposed to come from the factory free floating, check just to be sure before buying.
You don't have to stick with the big brands and new. Their are many good quality rifles out their and many of them used that will fit your budget and the owner maybe able to prove that the gun can shoot MOA or less rather than guessing with a virgin rifle.
As for the Savage hunter series it should do just fine. My question to you would be, "are you going to use this as a multipurpose rig, *hunt/target* and would a heavy barrel be an option?" As for the remingtons, they to, will do what you are looking for, but some of the lower end models might require some work such as free floating, and the trigger is not as nice as the accutrigger. Truth is most off the shelf guns can shoot moa or less with the appropriate small tweeks.The biggest 2 things to achieve what you are looking for is practice first and foremost, followed by ammunition preparation. As far as the practice part goes, given the fact you have been tinkering with milsurp rifles I would imagine you are at the point that you can shoot better than the cheap ammo will show. If not keep hammering away with the milsurp as it is cheap and readily available.
As for the caliber choice, 270 is a great round. The down side to that one is the selection of ammunition assuming that you are not handloading. The 'good stuff' is rather pricey and usually meant for hunting purposes. Although it will punch paper just fine, it can reduce your trigger time due to cost. Maybe look for something that has a wider bullet selection. This of course is all useless if you have the ability to load your own. If so, then go to town.