There is a reason that more local, state, and federal law enforcment agencies are using a Remington of one sort or another (generally something from their 700 series ) as their Swat Sniper Rifles.
What you say here is only PARTIALLY true. These agencies usually use Remington 700 ACTIONS, not rifles. I'd bet that 90%+ get some serious custom work done before they'll shoot well enough to be put into service. Most police SRT/SWAT sniper units require < 1MOA ALL THE TIME. Not average, but every single time, including the cold barrel shot. The Marine Corps uses the 700 as the basis for the M40 sniper rifle, but after it leaves the armorers at Quantico there's really not much Remington left in it.
As far as out-of-box accuracy...my nod would go to Savage. My Model 111 .30-06 with a 22" sporter barrel will keep right up with my Remington M700 VSS .22-250 with its 26" bull, with the Savage shooting factory ammo no less (haven't started handloading for it yet). Before I got the Rem VSS I had a Ruger M77MkII V/T that would also outshoot the Remington with the best handloads in both.
Savage's aren't the prettiest guns around, but they've got it where it counts.
The truth to the matter is that there are very few truly poor rifles made today, though a couple do come to mind. It's like a Ford vs Chevy vs Dodge thing. People tend to like one and hate another, but they all accomplish what they are designed to do. Get the rifle you like and pray it shoots well, as Savage and every other brand makes the occastional crappy 5 o'clock Friday rifle (meaning the last one off the line as the union whistle blows for the weekend). I can only say that in MY experience, my Savage's always have shot better than expected without any additional work. I look for accuracy and value, and out-of-box that's Savage. Now, if I were considering a basis for a custom rifle, with the intention to blueprint and rebarrel the rifle anyway, my first choice would be Remington.
So I think the best advice would be to look at all of them, then pick the one you like the best at the highest price you can afford to spend. Then put good glass on it and feed it the right ammo and it should do you well.
Good Luck,
Mike