Walking in the woods rifles I like to be light(er) weight. Even a shorty if it's over 12lb gets to be a bear after a while. That's one thing I would look out for.
Barrel quality makes precision. Factory Remington rifles are a gamble between 1/2 MOA and 4 MOA. Typically you can get into the 1 MOA or better range with the right ammo or reloads, though. Depending on the targets you intend to shoot, for 500-600yd a factory barrel could work fine. I'm just leery to say it "will work" because I've seen some pretty heinous Remington factory tubes.
IMO it's usually worth it to look at a Tikka T3. The actions are better (fit/finish), you get primary extraction, aftermarket is growing, and the barrels shoot, on the whole, better than Remington. MPA makes a chassis for the T3, as do many others now. Just something to consider, in my mind it's a no-brainer to take a Tikka T3 CTR (and have the barrel cut down) over almost any Remington unless the remington is a cheapo action donor to be built from the ground up.
Out to 500-600 you can go on JBM and play with different numbers for the different calibers. Pretty much anything will work, and be relatively equivalently hamstrung in the velocity department by the short barrel length. They will all work.