When rifles arrived in my shop with broken pull throughs stuck in the barrels, the variety of things to be removed was impressive and the failed attempts to solve the basic problem were quite imaginative. A tight patch and broken cord was rarely the only problem. Those are relatively easy to remove. The steel rods, nails, knitting needles were more challenging.
If the cord is broken right at the stuck patch, a close fitting rod can often push the blockage back the way it came. Not a dowel. I had a set of drill rods in different sizes that gave me a choice of the best rod to use. The rod is close fitting to prevent buckling. Polished steel to reduce the chances of marking the bore. Polished ends with rounded edges.
One notable failure that I will not forget was a .458 Ruger No.1. I forget now what all was stuck in the bore, but laddie had really had a go at it. I managed to clear the bore using a drill with a brass collar to prevent contact with the bore. Too late Laddie had already mangled the rifling. New barrel needed. Sent it to Ruger, they installed a new barrel and refinished the action. Cost $400 all in. Not bad I thought, given what a Ruger No.1 sold for. Barreled action was as good as new. Laddie stiffed me.