Depends on what type of 1000 yard shooting. F-Class, Palma, Service Rifle, etc.
Some limit what you can shoot. An example is NRA Service Rifle. You are limited to Service Rifles that shoot service rifle rounds, such as 223, 308, '06. In that case the 223s (ARs) are beating the M1s & M14s.
We all know that the the bigger bullet will buck the wind better (assuming proper bullets) then lighter bullets. That advantage is often offset by the recoil that comes with it. That is why there is the lighter bullets John mentioned. Great BC, little recoil.
I have two bolt guns I use in 1000 yards (non service rifle) matches. Both are Model 70 Winchesters, one in 300 WM and one in '06. As a general rule I do better with the '06 unless the wind is rally nasty, then the 300 works.
Just about any given 1000 yard match consist of 3 matches, (4 if you shoot the team match). Thats 3 X 20 rounds plus sighter's. I don't care how big and tough you are, the recoil will get to you. I would not even consider any thing about the 300 WM.
I like the military rounds, but there are many rounds out there that are much better, both in accuracy and recoil.