A321 ramp crew reported incident with tow bar during push back from gate.

Narrative

I was Crew Chief for departing Aircraft X. My crew of 3 fleet service employees consisted of Person A; Person B; and Person C. The flight was loaded; finalized; and ready to push out. Person A was the driver of the pushback tractor with connected towbar. Person B and Person C were the wing walkers. Person B walked airplane right; Person C walked airplane left. Person A pushed the plane out to the Aircraft Operations Area (AOA); turning the plane to properly follow the guidance line. I was observing my crew from the ramp near the jetbridge. Person A signaled to Person B that the brakes were set and the towbar could be disconnected. Person B removed the tow bar safety pin and released the locking mechanism/clamp. The towbar did not release or drop way from the nosegear as expected. It was clear to me that the there was tension on the towbar or it was not lined up perfectly straight with the plane. Person B shook the towbar back and forth in an effort to release it from the nosegear but it would not release. I watched as Person B walked to Person A to have a discussion. Person B then walked back to the towbar and tried to shake it again. After the first attempt to shake the towbar I began running out to the AOA to assist my crew. Person B and Person A had another brief discussion and Person A began pushing the plane rearward. The plane moved 4-5 feet rearward and stopped. The towbar was still attached to the nosegear and pushback. Person A then began to pull the plane forward. The plane moved 2-3 feet forward when the towbar disconnected and dropped. The plane continued to roll forward into the towbar head; sliding under the nosegear; wedging into the metal wheel of the nosegear on airplane right. The plane stopped forward movement when the towbar became wedged. I arrived at the plane just moments after the accident occurred. I quickly assessed the situation and noted the damage. I confirmed with Person A that the brakes were set on the plane and informed Person A and Person B that we would move the pushback tractor slightly rearward in order to release tension and disconnect the towbar from the tractor. Person A moved the pushback and I was able to disconnect the towbar from the pushback. Person A then moved the pushback so that I could remove the towbar from under the nosegear. I then took the headset and spoke with the Captain to inform him of what had happened. I then ran back to the jetbridge area to retrieve my radio to call operations and management.I watched the entire situation unfold.Cause: The event occurred because of a few crew errors. The first error was when Person A; the pushback driver; did not fully align the plane; towbar; and pushback tractor in a straight line on the AOA (aircraft operations area). From my perspective the towbar and airplane were in a straight line but the tractor was at a slight angle. This angle creates tension on the towbar head; making disconnection difficult. Second; after initially realizing the towbar was not disconnecting from the plane my crew should have stopped and asked for my assistance. Third; the correct solution to this problem should have been the disconnection of the towbar from the pushback tractor. Lastly; and most importantly the airplane should have never been moved with the towbar safety pin removed and head unlocked and open.Once the stuck towbar was removed from under the nosegear; I ran to get my radio and promptly contacted operations and management. Management arrived prompty and I was told to guide the plane back into the gate. Maintenance then inspected the nosegear.Suggestions: Ramp workers should immediately contact their crew chief with any issue before taking any action beyond the normal.