I have smoothed several production and WWI/II barrels using JB non embedding and more recently, their Bore polishing compound. JB uses a very find clay and the polish is like jewelers rouge. Follow the instructions and you will get a much smoother brighter bore.
Cleaning, however, may not be much better because copper build up in the machine marks BELOW the rifling surface. This stuff does not affect accuracy so I just don't bother with it. Only rough spots or burrs that can tear the jackets are of concern to me. If you run a dry clean cotton or felt patch down the bore, any above surface rough spots will catch some fluff. Looking down the bore will show you where the lint has hung up. That I would polish out. No issue with muzzle belling.
Butches bore shine is not designed to get rid of copper commonly found in production barrels. Works great in match barrels but then copper fouling is usually very low.
Best commercial stuff is Barnes CR10. Use outside, very high ammonia content. Will strip copper out of those machine grooves. Very quick to use too.
Ultimately, as long as the barrel shoots for a decent number of shots, how dirty it gets is irrelevant. Most production barrels need to have fouling shots anyway before they shoot their best. After all that hard work cleaning, two or three fouling shots and you are right back where you were before you cleaned.
I just shoot my rifles until accuracy fades. That is usually WAAAAAAYYY longer then the average person thinks.
My Savage in 270 has digested over 150 shots, starting to show some light blue streaks in the bore, still shoots 3/8" at 100yds if I don't screw up. What would cleaning possibly improve????
If you are having problems with solvents wetting the patches, the patches are made from treated cotton. Very common these days with low care fabrics. Just go get some cloth diapers or cotton fabric from Wally land and cut these up. Problem solved and less then 5% the cost of commercial patches if you don't count the bit of labour.
Jerry