Bullet weight has nothing to do with "longer range accuracy". I used 180 grain bullets in a 30/06 for many years until I discovered the bullets were spending more energy on the ground beyond the deer than inside the animal. That was also about the time I switched from a semiauto to a bolt action.
With the swap of rifle and ammo change to 150 grain, my area of domination expanded greatly.
As has been mentioned, the heavier bullets are usually more suitable for bigger animals and may not produce the same result when hitting deer. In the case of the 30/06, it has such an amount of power that it can waste some and still be king of the deer woods.
When I choose to use a 30/06 these days, my choice of bullet is the Nosler 165 grain Ballistic Tip. This is absolutely the best bullet(in the rifle I use) accuracy and performance wise that I've tried. I don't name my guns but my kids named my old 03A3 Springfield "The Buck Hammer" for obvious reasons. Powder burn to 400 yards, I haven't had to shoot a deer twice with this rifle in this century.