I never heard of break-in until I joined up at The Firing Line; the subject arose around 1999, I guess.
I can see why benchrest shooters would take such care. My own experience has been that accuracy of better than one Minute of Angle is mostly due to the bedding of the barrelled action in the stock, and some is due to tailoring handloads to the rifle.
Since my primary concern is hunting, and anything of one MOA is plenty good, I've never worried about the break-in process. If I'm only doing a sight-in or testing a new load, the relatively few shots mean to me that only an oily patch through the bore is sufficient. Every few boxes or so, I'll make a point of using a copper-removing bore cleaner.
My preference in a cleaning rod is a single-piece, 22-caliber stainless or teflon-coated stainless. I think it's best to clean from the chamber, to protect the crown of the muzzle. For casual cleaning, I've never had a problem with using WD 40 on a patch as a rust preventative and for a pass or two through the bore. (I don't spray directly onto the rifle.) Gun oil works just as well.
Art