I fail to see what a heavy rifle does better than a light rifle. I can take a factory Tikka t3 Lite, or whatever and make it group just as well as a heavy barreled 700 all day.
Your shooting at vitals... not for group size, so really why burden yourself with a heavier gun for something that should be 1 shot and done. Even IF you needed follow up shots (Sorry I can't relate, I was taught to shoot accurately and humanly on big game, so I don't shoot unless I am 100% sure.) A pencil barrel will still hold a group size that is certainly lethal.
NOW on the flip side of the coin. If someone is lacking in shooting fundamentals, a light weight rifle can make that persons mistakes more prevalent, (flinch, poor trigger control, ect.) but it is not the fault of the gun.