I've always cleaned mine by pulling the patches through from the chamber end to the muzzle end. I pull slow, I try to keep the rod centered in the muzzle and that removes the crud from the rifle/barrel rather than pushing it from the barrel into the chamber/receiver.
I don't clean the chamber, except with those same patches and I've not had "dirty chamber issues". Not with the M1A, AR15's, M1 Garand's or any hunting rifles.
Pushing a rod causes it to bend and rub the lands of the barrel. Pulling the rod will keep it much straighter and if you're careful, if won't make much, if any, contact with the lands.
Of course, using a jag means you have to push the jag/patch through the barrel, so you're going to get more rod to land contact. I never use jags, just the patch tip with patches/solvent or a brush, occasionally. If I shoot it, I clean it, that night, or the next morning, so my rifles/handguns have had a lot of rods/patches/brushes pulled through the barrels.