Is the ability to move a body part through a full range of motion OR
have control and the ability to lock that joint or body part in while doing work more critical?
Neither is more important.
We require a balanced combination of mobility and stability.
Too little mobility, just like too much mobility, can both result in a greater risk of injury.
Here's an easy example:
A stiff shoulder can result in an inability to lock out the arm in an overhead position. This will result in a less stable overhead position. And may compromise other areas of your body (like your back).
On the other hand, a hyper-mobile shoulder may present difficulty stabilizing the end range of the lockout or preventing excessive movement at the end range. This also results in a less stable overhead position.
Too much or too little of either compromises the way your body is intended to move and and raises the chance of you getting hurt.
Hope this helps!