I really don't believe that blowing down the barrel supplies enough moisture to soften the fouling very much. You will see the residual smoke from the ignition pouring out of the touch hole or nipple.
As far as reducing the size of the wiping jag goes, the jag can be reduced in steps of 0.010" until the diameter of the jag and the two thicknesses of the patch equal the land-to-land diameter. Now the damp patch will slide over the fouling in the grooves or the flash channel. As the jag and patch are pulled up, the patch bunches up and filles the grooves to pull the fouling from the grooves. The wiping patch should cover all the grooves of the wiping jag and about the same above the jag. Square or round patches make no difference. We still may have problems with a chambered breech, but the patch will push out the smoke to indicate a clear path from flame to powder charge.
When shooting, most of us are well aware whether the gun has fired or not. It is in situations where many guns are being fired simultaneously that we may not be aware if the gun has fired or not. Since most of us keep our eyes open when firing, we see the cloud of smoke even if the gun is heavy enough to prevent the feeling of recoil. It is very unlikely that as safety conscious as we are about the gun's loaded condition, that we are likely to be blowing down the barrel of a loaded gun. However, I do believe on a shooting range where several people are firing at one time, the restriction to eliminate blowing down the barrel is a reasonable safety precaution.
We cannot blow down the barrel as hard as we can move air when we push a damp patch down the barrel. I have done both and the smoke blown out is a slow, lazy plume of smoke. When I push a damp patch down the barrel after firing, the stream of smoke is blown far further from the touch hole.
Aside from the alleged safety issue, I just don't see an advantage in blowing down the barrel to wiping the bore to maintain consistent bore condition.