Speaking from a precision hunting perspective with 40+ years of experience:
The old Rem 700's factory rifles for the most part were good shooters. I have several that shot / shoot well under 1 moa, one in the 1/2 moa range (i.e. Varmint model) with tailored reloads. The old triggers were good, even very good, easily adjustable. But I'm afraid that era has past. The old days; if you bought a Rem 700 or Win 70 you could count on a quality rifle that would shoot very well. I only ever own one Win 70 and like a fool I sold it after reading gun magazine that said a 243 was not adequate for deer even though my experience was to the contrary. Young and foolish.
The Browning's I've owned (A and X bolts) shoot every well after the triggers were redone by a gun smith. My A bolt varmint stalker is a 1/2 moa rifle even with the marginal trigger. Don't think they make that model anymore - not surprising, if it works phase it out! That's what they must be teaching these days in business school! Reverse logic is in vogue.
The Tikka T3's, for the most part, have shot very well for me also, under 1 moa even better with some load development. These are good rifles worth considering.
But the absolute best, out of the box factory rifle I've ever owned is the Weatherby Supervarmint Master, Mark V. This rifle shoots well under 1/2 moa with tailored reloads. The trigger is very good. But it cost $1,500 dollars in 2009 dollars. Again, they don't make it anymore - not surprising - it works great so phase it out. It's my Number One Ghog Rifle.
Bottom line, with careful hand loading most factory rifle can be made to shoot in the 1 moa range or slightly under. If they don't shoot first address the bedding especially ones with the cheap plastic stocks. Also don't expect to shoot well if the trigger has a heavy, inconsistent pull or creep. If an after market trigger is available this something to consider after you address the bedding issue. Often free floating the barrel improves group size, at least that has been my experience.