My approach is almost identical to yours, Dave. A teakettle full of boiling rinse water, a little funnel in the muzzle, a glove and a towel gives me a scalding hot, rinsedbarrel. The patches and rod are right at hand, and I stuff adry patch down the bore as fast as I can afterward. First couple of patches show a lot of flash rust, sometimes almost black with the stuff. Goes away by the third or fourth dry patch, which come out almost pure white.

Then I coat the bore with either Bore Butter (patch comes out brown) or (lately) Sheath. Don't know whether the BB is picking up more flash rust, causing flash rust, or just turning brown from the heat.

On a stainless barrel, I don't see any brown after the rinse. Maybe a hint, if that, of gray. So I'm pretty sure, the boiling water rinse is causing flash rust.

Can't account for why you don't see it. Maybe something in the water here. Plenty of calcium, I know that.

I've been tempted to use only warm water, like Sproul, but, in all other respects, my cleaning techniques have worked well. That flash rust has got to be coming from somewhere and, over time, will probably take its toll.
Bob