Forget the Remmy
I think its the single shot Savage Model 12 FTR you should be looking at.
308 is great for out to 1000 yards - not so much with 223. 223 will get there, but it's not a recipe for long range success against a 308. At 300 yards yes, 600 yards maybe, 1000 yards I don't like your odds.
For long range shooting you need to look closely at the action when it's out of the stock.
You want to examine how much steel there is in the middle of the action and over the ejection port. The web of steel across the top of the Savage helps support the weight of the barrel and contributes to accuracy. Stiff is the name of the game. How stiff you ask? Well stiffer than a Remmy I can tell you that. (unless its a single shot 40XB)
A Rem 700 is cleared top and bottom and for that reason is nowhere near as stiff as the Savage Model 12 action.
I've seem guys compete side by side with Barnards and Savages and as often as not the guy shooting the humble Savage wins.
A rifle is only as good as its weakest link and if you use a Remmy that
will be it.