This has been an enlightening post. I have a new (never fired) SA M1A. I called SA CS regarding whether I needed to do this, or not. They also recommended the use of a bore snake. They referred me to the Krieger website. The entire article is attached.
"Because the lay of the finish is in the direction of the bullet travel, very little is done to the bore during break-in, but the throat is another story. When your barrel is chambered, by necessity there are reamer marks left in the throat that are across the lands, i.e. across the direction of the bullet travel. In a new barrel they are very distinct; much like the teeth on a very fine file. When the bullet is forced into the throat, copper dust is removed from the jacket material and released into the gas which at this temperature and pressure is actually a plasma. The copper dust is vaporized in this plasma and is carried down the barrel. As the gas expands and cools, the copper comes out of suspension and is deposited in the bore. This makes it appear as if the source of the fouling is the bore when it is actually for the most part the new throat. If this copper is allowed to stay in the bore, and subsequent bullets and deposits are fired over it, copper which adheres well to itself, will build up quickly and may be difficult to remove later. So when we break in a barrel, our goal is to get the throat "polished" without allowing copper to build up in the bore. This is the reasoning for the "fire-one-shot-and-clean" procedure."
I believe that Winchester made a similar recommendation to me.
Update - I just contacted SA CS. While they do use chrome moly barrels, they are not Krieger barrels. However, the information in the attached file from Krieger explains the process and is easy to follow. SA recommends 2-3 rounds to heat up the barrel, run the bore snake from the breach to the muzzle, then run new cotton patches to check that the copper fouling has been removed. If dirty, repeat the process. If clean, continue firing.