Been shooting sub-MOA with centerfire since 1950. Never heard of "barrel break-in" until coming to TFL in 1998.
According to what Gale McMillan posted here, back a dozen or so years, one of his competitors in the barrel-making business dreamed it up as a way to get guys to shoot more, early on, and burn the leade faster than otherwise. This would let him sell more barrels to the bench-rest crowd. They're picky about group size, right?
So somebody hears guys talking about this wondrous methodology, and repeats it here and there. Some youngun takes it to heart, grows up and goes to work for Remchester and talks about barrel break-in. Next thing you know, it's in their instructions. So then somebody from Savage reads it and away we go.
Then comes the Internet and Oh. My. Gawd. "Everybody knows."