So, you link to an Eley article and give no summary of it's contents and how they relate to, or support your cleaning methods. I've copied what I assume you are using to support your position out of the article:
The barrel should be cleaned with a bronze brush. In order to do this you will need to push the brush in the direction of the muzzle and repeat three times. A bronze brush is used as it is the only tool that can be used to completely remove lead remnants.
Wow, only three times? Now your rifle is operating room clean and you may go about your merry business?
The cleaning there only applies to high quality, match grade hand lapped or otherwise finished barrels (Anschütz, etc..). I can tell you from personal experience with a Lilja barrel and Anschütz barrels, as a borescope owner, that brushing these barrels is seldom necessary. Usually, they clean right up with regular patches and solvent. The carbon ring in the chamber may take some more work, depending on how badly you've let it develop. I'm finding a .270 cal nylon brush short stroked in the chamber is doing a very good job of cleaning out moderate carbon rings. Severe cases require JB bore paste. They do not mention removing the brush for the back stroke, but it is implied. No statement to support any reasoning for removing the brush. Not a good source.
There is no cleaning regime of "X" quantity patches with, and without solvent plus brushing with either brush material for "X" quantity strokes that will guarantee you a clean barrel every time.
Every barrel is different in it's needs, and each shooting session usually is not the same. What type of ammo did you shoot, how many rounds? etc. Do you stay on top of preventing carbon ring development, or has this been building for years? These patchworm and light cleaning regimes suggested
will certainly come back and bite your accuracy down the road. Hoppes #9, a bronze brush and patches are not 100% effective, especially for the carbon ring. Once I got my borescope and checked out my 16 year old Remington 597 that was cleaned this way for it's lifetime, I saw how badly fouling had built up and gave it a deep cleaning with JB bore paste. Previously, it was a 1" plus shooter at 50 yards, cleaning brought it back under 1" (0.6"-0.8" typical) and even tosses me some sub 1/2" groups. I know of no rifle that shoots it's best heavily fouled.
The only way to determine the effectiveness of your cleaning is borescope inspection. If you find over the years accuracy isn't what it used to be, this suggests it may be time for a deep cleaning with JB bore paste. Above all else, keep the chamber/throat area as clean as possible, maintain regularly after each session. Production grade barrels require much more aggressive cleaning than highly finished target barrels. If we're competing against each other, disregard everything I'm saying