Navy LCDR - It's not the rifled bore that benefits, but before the rifling begins.
According to rifle manufacturers like Dan Lilja, here's the answer:
The rifling itself isn't made by spinning a tool inside the bore, so that's not the problem.
But the chamber and freebore leade (the unrifled protion just beyond the chamber) are cut with reamers that are rotated inside the bore. That leaves concentric cut marks PERPENDICULAR to the travel of the bullet. Those first few centimeters are absolutely vital to a bullet centering itself in the bore so it cuts into the rifling correctly. It is truly vital because whatever attitude the bullet takes in the bore is the attitude it will take in flight.
A high-dollar custom barrel is carefully hand-lapped in that area to remove the tool marks. So a really really nice high-dollar rifle probably doesn't need it. But most barrels (especially production barrels) are not hand lapped.
So 'break in' is a procedure to let the first 10 to 50 rounds do that lapping for you. Be mindful that the freebore leade is the most critically vital, barrel makers tell us that the bore itself is important but less vital.
As the first 10 to 50 rounds are fired, the guilding metal brass jacket does that honing job. But to be effective you must remove the copper constantly, or the successive bullet will just ride on the copper deposit without lapping the machine marks. Most decent barrels will smooth the cut marks in 10 to 12 rounds. If really bad (a cheapie cheapie barrel) it may take as many as 50 rounds.
So how much difference does that make? If you blast factory ammo at targets for fun, you won't tell the difference. If you hunt game at reasonable distances (say 300 yards or less) you probably won't tell the difference. Those who start with a normal production gun and want to get the very most will likely enjoy a benefit. How much benefit? It depends upon how bad the cut marks were. And that varies individually from one barrel to the next, even on identical rifles made on the exact same production line just weeks apart.
Of course, this individuality makes it nearly impossible to quantify your own benefit. If you happen to take possession of a rifle with very few cut marks in the freebore, and an excellent bore all the way to the muzzle crown, it probably shoots great anyway. If you happen to get a lemon with a totally crap barrel that can't hit the wall of a barn from the inside (with the doors closed) all the break-in in the world won't help it.
You determine how important this issue is to you, and you make your own decision. It's your gun, not ours. You decide what makes you happy.
NOTE: The critical procedure is simple. Clean the barrel between shots for those first 10 or 12 rounds. Nothing more. No special ammo, no special time of day, no special prayer before each shot. Take the first dozen shots just as you would with any new rifle, setting the sights and getting accustomed to shooting it. Just take your cleaning kit with Copper bore cleaner (standard Hoppe's won't disolve copper from the cut marks) and clean between shots.
