Okay, so we don't start a fight let me just say this... Whether you believe in 'break-in' or not. It wont hurt anything, assuming you clean correctly and don't damage anything. It wont reduce barrel life, and it wont break anything.
Here's my suggestion. Take 1 or two shots, just to make sure the gun functions as intended. Clean between each shot. Now mount your optics, and begin getting your zero, cleaning between each shot. Eventually you'll want to test groups, and see how your new barrel groups. Shoot 3 shot groups and then clean. Repeat this a few times, using different ammo, to get a feel for how it shoots. There, that's 1/4 to 1/2 of 'break-in' and no ammo or barrel life was wasted. The most you have to lose is a couple dollars worth of cleaning supplies, and some free time.
1) They think it improves accuracy 2) They think it results in a barrel that holds less copper per given shot string.
I don't believe #1 to be true, but #2 can CERTAINLY hold true, in certain barrels. That said, a high quality match grade barrel, likely will not benefit as much as a lower grade hunting barrel. IMO the only way 'break-in' would affect accuracy directly, is if a burr of imperfection was stripping enough copper off each bullet, to throw it off balance. A broken in barrel MAY stay accurate for more shots, because it should be accumulating less copper fouling, which is known to degrade accuracy, over time, until you clean it out.