I'm curious why there's a need to cut the tubing. From the diagram:
Fresh food section replacement parts for General Electric PSC23MGPAWW Refrigerator | AppliancePartsPros.com
it appears the water tank (810) and water line (812) are 2 separate parts. I assume there's a way to remove the tubing from the water tank and re-use the old tubing. So, I'm not sure why there's a need to cut the tubing and reconnect with the quick connect.
Also, do you absolutely need to turn off water supply at the back of the fridge? I only see water running through the tube when the water dispenser is activated/pushed.