Hi niina, the behaviour you are seeing without the head in place is what I would expect. So I do not think there is anything wrong with the counterbalancing.
The cable you see hanging down inside the column should pass through a hole in the lens carriage, and it should have a cylindrical fitment on the end. The fitment is too large to go through the hole in the lens carriage, so when the head is raised it reaches a point where it pulls the lens carriage up as well, preventing the bellows being overstretched. It sounds as if this part is missing on your enlarger.