Here is another opinion that is worth what your paying for it.
When you start to pull the patch and jag out it is not the bunching up on the grooved part of the jag that is the problem but the bunching up on the shoulder above the grooved part.
Try using a jag with no shoulder and a very slender cleaning rod, AKA a modern Dewey or Pro-shot rod. and see if that solves your problem.
Also, in case it wasn't mentioned, once it is stuck, squirt some thin lubricant down and let it soak into the patch before trying to pull it again, Kroil or similar.