All depends on the barrel. My best barrel, a PacNor 3-groove, has never seen a brush of any kind, only soaking wet patches and Wipe-Out. I have never had a carbon ring, no stubborn copper fouling. Throat has advanced only .007 in 800 rounds, crown is like new. Velocity is within 8 fps of load when brand new. Accuracy is same as new, if not better.
Regarding pulling a brush off after out-stroke: 1) By "sawing" the brush both directions, you are forcing the bristles to change direction right over the most delicate area of your barrel; 2) You are dragging the same residues back into the bore that you are trying to get rid of; 3) Removing the brush allows you to clean and flush the crap off the brush before you send it back through the barrel breech to muzzle -- this takes only seconds.
I have viewed the crown of barrels that got a lot of 2-way back and forth brushing with a bronze brush. The edge of the crown looked like shark's teeth. Both Tim North (Broughton) and John Krieger (Krieger) advise against dragging a bronze brush back across the crown. Why saw your brush both ways if there is no benefit and potential negatives?
I advise folks to use the least invasive methods to clean bores. Yes some of the short-range benchrest guys often brush "early and often". But they are also prepared to recrown and replace barrels after as little as 600 rounds.
WyleWD wrote: Personally, my experience is that the brushes or the twisted wire makes very tiny gouges in the bore, thus allowing for build up of carbon or copper or both in those minute scratches.
-- I can't confirm this scientifically one way or another, but I think there may be some truth to it. I can only say that with soaking wet patches, and Wipe-out I have zero carbon problem, zero copper fouling problem, and that's confirmed with Borescope. As far as pressure issues, as noted above, my measured velocity has pretty much stayed stable.
Other barrels may require brushing from time to time. But I think many shooters still attack their barrels much too aggressively. And I regularly see shooters bow their rods while cleaning, or fail to use a good cleaning rod guide.
BORESCOPING
This can be an important tool but don't be lured into the obsessive pursuit of a mirror-like internal finishes. Many barrels shoot best slightly fouled. AND, if you scrub your bore with abrasives obsessively just to give it a pleasing appearance for your borescope, you may end up hurting the barrel's accuracy. As they instruct doctors, the first rule of barrel cleaning should be DO NO HARM.