It sounds like a center rotation pin issue. I noticed on my old cobalt when I pulled it up and to the left like I always did, after about 4 years, if I tried to put it down straight the round pin in the middle (which for me was a round grommet or solid pin made from a small piece of rolled steel) was catching on the bracket in between the two halves of the of the handbrake where the cable rolls, and it was bonded to the other part of the assembly, so it would catch, and get stuck. I am assuming your button will physically go inward if you press it without any issue, but the assembly won't roll downward towards the console like it should. Also something that can happen is with temperature, the wheel the wire presses on in the handle can get warped a couple thousandths of an inch with time and heat/repeated angled usages and go in front of the bracket that bolts to the body and prevent it front going downward as well.
Check to make sure the wheel with the wire on it isn't getting stuck on anything and the center pin that allows the handle to rotate from the body bolted assembly is lubed.
It is probably not your cable, as it would just push out and curl out the end of the cable near the calipers without much excess force. I would try removing the center console around the handbrake, remove the plastic cover, throw some spray lithium lube in there around that center pin and see if that fixes it. If it helps but still resists, you'll know it's the handbrake assembly itself that needs to be replaced.